\section[]{Introduction}
In the last few years a new generation of models for the chemical evolution
of the Galaxy has started to appear, in which also the dynamics of the system
is taken into account (e.g. Sommer-Larsen \& Yoshii 1990, Chamcham
\& Tayler 1994, Hensler, this volume, and references therein).
This new class of models can provide a complete
scenario for the evolution of the Milky Way but is still in a rather
preliminary phase. The aim of this
presentation is then to review the current state of {\it standard}
models for the chemical evolution of the galactic disk, with particular
emphasis on the effect of gas accretion on the element abundances and
gradients.

These models are quite successful in accounting for the large scale - long
term phenomena taking place in the Galaxy, and reproduce its major observed
features, such as the age-metallicity relation and the G-dwarf distribution
in the solar neighbourhood, the elemental and isotopic abundances and ratios,
the present star formation rate, gas and total mass densities. To avoid
misleading conclusions, it is however necessary to test the models
by comparing their predictions not only with the observational constraints
derived for the solar neighbourhood but also with the data relative to other
galactic regions; first of all because the solar neighbourhood is not
representative of the whole disk, and secondly because the distribution with
galactocentric distance of several quantities provides important information
on the history of the Milky Way.

The current rate of star formation (SFR) represents an excellent example of the
importance of modeling the whole disk. One of the most popular approximations
for the SFR is a linear proportionality with the gas density, and some authors
consider it not only a simple and intuitive law, but also a realistic one
because it can reproduce several properties observed in the solar neighbourhood
objects. Since the observed radial distribution of the gas in the disk is
rather flat (see the shaded area in the bottom panel of Fig.2),
this approximation inevitably implies a flat radial distribution of
the present SFR. However, Lacey \& Fall (1985) have shown that the current
SFR in the disk, derived from a large sample of young objects (pulsars,
O-stars,
etc.), is actually a steep decreasing function of the galactocentric distance.
Thus, the radial distribution predicted by models assuming a SFR linearly
proportional to the gas density is totally inconsistent with the observed one.
On the contrary, models assuming the SFR proportional to both the gas and
the total mass densities (e.g. Tosi 1982 and 1988, Matteucci \& Fran\c cois
1989,
hereinafter MF)
are in agreement with the observed trend, thanks to the steep decrease of
the total mass density with galactic radius.

\section[]{Gas infall and abundance gradients}

The idea of a long-lasting infall of metal poor gas on the galactic disk
was first suggested by Larson (1972) and Lynden-Bell (1975) to solve several
inconsistencies of the first simple models: a too rapid gas consumption that
prevented to reproduce the amount of gas currently observed in the disk,
the unlikelihood of a complete collapse of the whole protogalactic
halo in a few 10$^8$ yr, and the existence of very few metal poor, long
living stars in the solar neighbourhood, compared to the relatively large
predicted percentage (the so called G-dwarf problem). Since then,
gas accretion has turned out to be necessary to explain most of the
characteristics of our disk, and all the chemical evolution models (with or
without dynamics) in better agreement with the largest set of observational
constraints assume a significant amount of gas infall throughout the
disk lifetime.

\subsection[]{Interstellar medium abundances}

One of the most evident effects of gas infall on galactic evolution concerns
the absolute value and the radial distribution of the element abundances.
If no gas accretion is assumed after the disk formation, all chemical evolution
models with reasonable SFR and initial mass function (IMF) predict too large
present abundances and/or inconsistent radial distributions. This problem is
apparent in the top panel of Fig.1 where the distribution of the current
oxygen abundances predicted by models with exponentially decreasing SFR and
Tinsley's (1980) IMF is compared with that derived by Peimbert (1979) and
Shaver \mbox{\it et al.} (1983) from HII regions observations. The models divide the disk
in concentric rings, 1 kpc wide, and assume the sun at 8 kpc. Gas motion can
be allowed between consecutive rings, whereas stars are assumed to die in the
same region where they were born. If the galactocentric distances of the
HII regions are properly rescaled assuming the sun at R=8 kpc,
the observational oxygen gradient is
$\Delta$log(O/H)/$\Delta$R = -0.103 dex kpc$^{-1}$. The long-dashed line
corresponds to a model with no infall after the disk formation 13 Gyr ago: it
is too flat and lies above the observed range of oxygen abundances. The solid
line, instead, fits very well the data and corresponds to a model with SFR
e-folding time of 15 Gyr and constant infall of primordial gas with density
rate F = 4 10$^{-3}$ M$_{\odot}$kpc$^{-2}$yr$^{-1}$ all over the disk.
This uniform density rate implies a larger mass of metal free gas infalling
in the outer than in the inner rings and favours the development of negative
metallicity gradients as steep as observed.
This model reproduces the most important features of the Milky Way and from
now on it will be referred to as the {\it reference} model. It must be
emphasized, however, that other combinations of the SFR and infall
parameters may lead to a similarly good agreement with the data, as
shown in Fig.1 by the short-dashed line, corresponding to a model assuming a
shorter e-folding time for the SFR (5 Gyr) and a lower infall density rate
(F = 2 10$^{-3}$ M$_{\odot}$kpc$^{-2}$yr$^{-1}$).
What is generally found is that models in better agreement with the
observational constraints assume SFR e-folding times in the range 5-15 Gyr
and e-folding times for the gas accretion rate longer than for the SFR.
This requirement
is not unrealistic, if we consider that according to Sofue's (1994) models the
Magellanic Stream is regularly supplying the Milky Way with gas since 10 Gyr.

\begin{figure}
  \vspace{9truecm}
  \caption{Present distribution of the oxygen abundance with galactocentric
  distance, as derived from HII region observations (dots) and theoretical
  models. The average observational uncertainty is shown in the bottom left
  corner. Top panel: models with primordial infall (see text for details).
  Bottom panel: models with metal enriched infall.}
\end{figure}

If the mass of infalling gas is assumed to increase inwards rather than
outwards the model predictions are much less satisfactory. The dotted line
corresponds to a model with infall rate proportional to the total mass of
each ring, which is then increasing toward the center. As a consequence,
there is a larger dilution of the inner interstellar medium (ISM) resulting
in a flat abundance distribution inconsistent with that derived from
HII regions.

The above results refer to infalling gas with primordial chemical composition,
which would be available at best in the intergalactic medium or in the early
halo. If the gas originates from regions already polluted by stellar
nucleosynthesis, such as the current halo or the Magellanic Stream,
it has most probably a non negligible metal content. The intermediate solid
line in the bottom panel of Fig.1 shows that if the
metallicity of the infalling gas is 0.5 Z$_{\odot}$ the predictions
of the {\it reference} model are at the upper edge of the observed
distribution.
The top solid line shows that if the infall metallicity is solar the resulting
oxygen abundance is definitely outside the observational range. From a large
variety of models, Tosi (1988b) has found
that to allow for a good agreement with the data the infall metallicity
should not exceed 0.3 Z$_{\odot}$. The same result has been obtained by
MF with different models and different assumptions on the relative abundances
of the infalling elements. This limit is perfectly consistent with
the metal content attributable to both the galactic halo and the Magellanic
Clouds.

Depending on the model parameters, the present infall rate for the whole
disk ranges between 0.3 and 1.8 M$_{\odot}$yr$^{-1}$. The lower limit of this
range is in agreement with the amount inferred from Very High Velocity Clouds
which, with the Magellanic Stream, are the most reliable observational
evidence for this phenomenon. If a fraction of High Velocity Clouds
could be considered of non disk origin as well (see Danly, this volume),
the amount of infalling gas observationally detected would cover all the
theoretical range.

\begin{figure}
  \vspace{9truecm}
  \caption{Radial distribution at three different epochs of quantities
   predicted by the {\it reference} model. Top panel: SF and infall rates;
   bottom panel: gas and total mass. The shaded area corresponds to the
   observed gas mass range as published by Lacey \& Fall (1985)}
\end{figure}

The metallicity gradients predicted by chemical evolution models depend on
the ratio between the SFR and the interstellar and infall gas predicted at
each epoch and at each galactocentric distance. The top panel of Fig.2 shows
the radial distribution of the SFR and infall rate resulting from the
{\it reference} model at three different epochs: the dotted line corresponds
to the epoch of disk formation (assumed to be 13 Gyr ago), the dashed line
to the epoch of sun formation (8.5 Gyr later), and the solid line to the
present. Since the infall rate is assumed to be constant,
only one line appears in the
figure. The bottom panel of Fig.2 displays the radial distributions of the
gas and total mass at the same three epochs. The disk is supposed
to evolve from an initial configuration of pure gas with radially decreasing
mass (dotted line). The initial SFR is radially decreasing as well, so
that in inner regions there is more astration and therefore larger stellar
production of metals. However, the amount of ISM gas which must be polluted
by these metals is much larger in the inner regions and therefore
the efficiency of the ISM enrichment is quite modest. Thus, at early epochs
the predicted abundance gradients are either flat or even positive, depending
on the model parameters
(see also Moll\'a \mbox{\it et al.} 1990). After several Gyr, the situation changes
significantly, because the larger astration of the inner regions leads to
a higher gas consumption which is not totally compensated by infall
since the gas accretion is assumed to be increasing outwards. Thus, at a
certain time (see for instance the dashed line corresponding to the situation
4.5 Gyr ago) the larger SFR in the inner regions corresponds to a higher
efficiency in the metal enrichment of the medium and negative abundance
gradients start to develop. Since then, and as long as the star formation
activity remains a decreasing function of the galactocentric distance,
the slope of the gradients keeps steepening, because the gas radial
distribution
becomes increasingly flat, the infall dilution is more efficient outside, and
the metal enrichment inside. As shown in Fig.2, the disk of the Galaxy is
currently in this phase with a very flat radial distribution of the gas mass,
a radially decreasing SFR and a SFR/infall rate ratio progressively smaller
for increasing R and equal to 1 at 9-10 kpc (see also Wilson \& Matteucci
1992).

A steepening with time of the abundance gradients is predicted by most
of the models (with or without dynamics) which are able to reproduce the
observational features of the Galaxy (e.g. MF, Chamcham \& Tayler 1994,
Koppen 1994) despite the rather different model characteristics (see, however,
Ferrini \mbox{\it et al.} 1994 for different predictions). It is
then important to verify if this predicted trend is indeed
consistent with the available observational constraints. Since the gradient
derived from data on HII regions is representative of the situation in
the current ISM, to test the model predictions older objects must be
examined as well.

\begin{figure}
  \vspace{9truecm}
  \caption{Top panel: Radial distribution of the oxygen abundances 3 Gyr ago
   derived from the {\it reference} model (solid line) and from observations
   of PNeII (dots). The data are from PP and the dotted line represent their
   best fit. Bottom panel: same, but for the He abundance.}
\end{figure}

Planetary Nebulae, specially of Peimbert's (Peimbert \& Torres-Peimbert 1983)
type II (PNeII), are also
very good indicators of the ISM metallicity. PNeII have stellar progenitors
with lifetimes in the range 1-5 Gyr and therefore represent the ISM
conditions around 3 Gyr ago. Two recent and extensive studies (Pasquali
\& Perinotto 1993, hereinafter PP, and Maciel \& Koppen 1994) show that the
abundance gradients derived for several elements in PNeII are systematically
flatter than the corresponding gradients derived from HII regions. For
instance,
the oxygen gradient derived by PP is
$\Delta$log(O/H)/$\Delta$R = -0.03$\pm$0.01 dex kpc$^{-1}$ and that derived
by Maciel \& Koppen is -0.07$\pm$0.01. The latter authors have also found
hints of increasing slopes of the gradients with decreasing age of the PNe
(i.e. from type III to type I), in agreement with the model predictions.
Fig.3 shows the helium (bottom) and oxygen (top) abundances as derived by
PP from PNeII and the corresponding predictions of the {\it reference} model.
The agreement between the model solid line and the empirical best fit to the
data (dotted line) is excellent. This confirms that in the last few billion
years the slope of the abundance gradients in the ISM has actually steepened.

No other gaseous indicators are available to check whether the gradients were
increasingly flatter at earlier epochs. Stars and stellar clusters of
whatever age are instead visible in a fairly large range of distances
and can therefore indicate what was the earlier scenario.

\subsection[]{Stellar abundances}

As far as single stars are concerned, the situation is unfortunately rather
confuse. Lewis \& Freeman (1989) found no significant metallicity gradient
in a sample of 600 old K-giants, but more recently Edvardsson \mbox{\it et al.} (1993)
have argued that the radial metallicity distribution derived from
a sample of 189 F and G-dwarfs is similar to that derived from HII
regions. The major result of this accurate and extensive work is the
scatter on the derived abundances which turns out to be much larger than
the observational uncertainties and should then be considered an intrinsic
feature of the analysed stellar population. Edvardsson \mbox{\it et al.} therefore
avoided to formally derive the slopes of the abundance gradients, but one
can obtain them from their table 14 where the analysed stars are divided in
different groups according to their age and galactocentric distance and the
average [Fe/H] of each group is given. Despite the poorness of the sample
in the older and more distant bins, and the corresponding weakness of the
statistics, it is interesting to point out that the resulting formal slopes
get flatter for increasing age (i.e. toward earlier epochs) and that the
oldest bin even shows a positive gradient (derived from two single points,
however !), thus giving some further support to the predictions of the
chemical evolution models.

\begin{figure}
  \vspace{6truecm}
  \caption{Age-metallicity distribution for $\alpha$ elements as derived from
  the {\it reference}   model and from Edvardsson \mbox{\it et al.} (1993) data. Filled
  circles and dashed curve refer to stars in rings with 4$\leq$R$\leq$7 kpc,
  open circles and solid curve to stars with 7$\leq$R$\leq$9 kpc, crosses and
  dotted line to stars with 9$\leq$R$\leq$11 kpc. Age is in Gyr.}
\end{figure}

Another interesting feature of the Edvardsson \mbox{\it et al.} data is the different
distribution of metallicity with age for stars at different galactic locations.
As pointed out by Pagel (1994) and shown in Fig.4, if one divides their stars
into three groups according to their mean galactocentric distances (inner
objects, solar ring objects, and outer ones), one finds
that the outer stars show a much flatter age-metallicity distribution,
with average abundances in the last ten billion years (i.e. over most of the
disk lifetime) systematically lower that those of the other objects. As already
mentioned above, the data show a large intrinsic scatter in the derived
metallicities of stars of any age and this scatter cannot be directly
reproduced by standard chemical evolution models like the {\it reference}
one which assumes both the SFR and the gas accretion in a sort of steady-state.
However, the age-metallicity relations predicted for the three ranges of
galactocentric distances
are consistent with each of the corresponding average empirical
distributions. Notice that the relations predicted for the solar (solid
line) and the outer (dotted line) rings flatten off at recent epochs, whereas
the relation predicted for the inner ring (dashed line) keeps increasing
up to the present time, as already shown by MF. Besides, Fran\c cois \&
Matteucci (1993) have argued that even the spread of the observed
age-metallicity distribution can be accounted for by {\it standard} models,
once the different birthplaces of the sample stars are considered.

The major problem of single stars analyses is the uncertainty in the derived
R, age and metallicity of objects beyond a limited distance from the sun
as confirmed by the Edvardsson's \mbox{\it et al.} survey. From this point of view,
open clusters are in principle safer indicators. There are, however, several
problems affecting also the determination of the cluster parameters, such as
the non homogeneity of most age estimates and the uncertainty on the cluster
original birthplace, the cluster disruption due to disk friction which can
alter the original distributions, etc. (see, however, Carraro \& Chiosi 1994).

Several years ago, young open clusters have been suggested to indicate steeper
abundance gradients than old open clusters (Mayor 1976, Panagia \& Tosi 1981).
However, more recent and extensive studies (Friel \& Janes 1993, Thogersen
\mbox{\it et al.} 1993) do not seem to support this hypothesis. The metallicity gradient
derived by Janes and collaborators is
$\Delta$[Fe/H]/$\Delta$R = -0.09$\pm$0.02 dex kpc$^{-1}$
for the whole sample of clusters of any age, and does not seem to depend on
the cluster age. Bearing in mind that for field stars in the disk oxygen has
been empirically found to follow the relation [O/Fe]$\simeq-$0.3[Fe/H] (e.g.
Edvardsson \mbox{\it et al.} 1993), and assuming that this relation applies to open
clusters
as well, the iron gradient corresponds to
an oxygen gradient $\Delta$log(O/H)/$\Delta$R = -0.06, flatter indeed than
that derived from HII regions and more similar to that of objects (as the
PNeII discussed above) a few Gyr old.

\begin{figure}
  \vspace{5.5truecm}
  \caption{Radial distribution of open clusters metallicity as derived from
  Friel \& Janes (1993) and Thogersen \mbox{\it et al.} (1993) samples. The clusters have
  been divided in age bins and the linear best fit for each bin is shown
  (dotted line for the oldest bin, long-dashed for the 4-5 Gyr bin,
short-dashed
  for the 2-4 Gyr bin, and solid line for the youngest one.}
\end{figure}

The most striking feature of Friel \& Janes' sample is that at each galactic
radius the oldest clusters are also the most metal rich (see Fig.5). It is true
that the published clusters in the oldest age bin ($\geq$8 Gyr) are only four
and the corresponding statistics is therefore too poor; however
two additional clusters of roughly the same age have been found (Friel 1994,
private communication) with less extreme metal abundances but still higher
than average. It is of crucial importance to verify this result
with a larger sample of old clusters and with more accurate and
homogeneous methods to derive their ages, chemical abundances and galactic
original locations. It might well be, in fact, that this anomaly is fictitious
and resulting from the uncertainty in the metallicity and/or, more probably,
age determination. However, if confirmed, this phenomenon would have remarkable
implications on our understanding of the Galaxy evolution, because it is
opposite to any intuitive age-metallicity relation derivable from {\it
steady-state} scenarios where old stars are inevitably more metal poor than
young objects and may result from short, intense phenomena not considered in
our models.

Another characteristics of old open clusters is that all of them are located
beyond 7-7.5 kpc from the galactic center, contrary to younger clusters
which are likely to be equally distributed everywhere in the disk (Janes
\& Phelps 1994). On the one hand, the external location of the older clusters
in the observed sample
might not reflect an odd distribution of all the clusters formed several Gyr
ago and be the result of a more efficient disruption in the inner than in the
outer regions. On the other hand, it may instead correspond to a non
homogeneous
star formation activity, perhaps related to external phenomena like the first
impact on the disk of the Magellanic Stream (Sofue 1994). The latter
scenario might also provide an explanation to the large metallicity of the
oldest clusters, in terms of a transitory metal enhancement of the ISM due
to the larger SFR triggered by the sudden event and later smeared out during
the following {\it steady-state} evolution.

\section[]{Summary}

In conclusion, the comparison between the abundance distributions in the
galactic disk predicted by {\it standard} chemical evolution models and
derived from observational data can be summarized as follows:\\
a) Abundance distributions derived from observations of gaseous objects of
various ages (HII regions and PNe) are very well reproduced by {\it
steady-state} models with slowly decreasing SFR and large - long lasting
infall of metal poor gas.\\
b) Average abundance distributions derived from stars are also reproduced.
These {\it standard} models, however, do not reproduce the observed spread
in the metallicity distribution of field stars and the anomalously high
[Fe/H] of the oldest open clusters.

We must bear in mind, however, that old stars can have quite eccentric orbits
and can therefore have formed in galactic regions different from those where
they are observed now. According to Fran\c cois \& Matteucci this might explain
most of the abundance spread in the Edvardsson's \mbox{\it et al.} sample. On the other
hand, Carraro \& Chiosi (1994) have suggested that the same argument cannot
apply to the case of old open clusters, for which other explanations are
thus needed, unless all the inconsistencies with the model predictions can
be attributed to observational errors.

A possible reason for the different agreement found for gas and stellar
objects is that the gas mixes rapidly, compared to the timescales for
galactic evolution, therefore forgets local perturbations occurred in the
past and follows the {\it steady-state} scenario. Stars, instead, keep
memory of the local perturbations occurring at, or just before, their birth
and therefore deviate more from that scenario, showing intrinsic large
scatter and anomalous behaviours. To interpret in detail their observed
features more sophisticated models taking into account also the small scale,
short term phenomena are therefore required.

\vskip 1truecm
I wish to thank Francesca Matteucci for always being ready to discuss and
compare the results and the different approaches of our models: a
praiseworthy attitude rather unusual among theoreticians.


\section{Introduction}
Graphs considered in this paper are finite, undirected and simple.
For a graph $G$, let $V(G)$ be the vertex set of $G$ and $E(G)$ be the edge set of $G$.
Let $v(G)$ denote the order of $G$, i.e., $|V(G)|=v(G)$.
For $u\in V(G)$,  $N(u)=\{v: uv\in E(G)\}$, $N[u]=\{u\}\cup N(u)$ and $d(u)=|N(u)|$.
Denote the chromatic number of $G$ by $\chi(G)$, and let $s(G)$ denote the chromatic surplus of $G$, which is the number of vertices in a minimum  color class over all proper $\chi(G)$-colorings of $V(G)$.
Let $U$ be a subset of $V(G)$.
Denote $G-U$ as the graph obtained from $G$ by deleting $U$ and all edges incident to $U$.
Denote a complete graph on $n$ vertices by $K_n$, and a path on $n$ vertices by $P_n$. The union of $k$ disjoint copies of a graph $F$ is denoted by $kF$ and the disjoint union of $G$ and $H$ is denoted by $G\cup H$. The join graph of $G$ and $H$, denoted by $G\vee H$, is  the graph obtained from the vertex disjoint union $G\cup H$ by joining each vertex of $G$ to each vertex of $H$.

Let $G$ and $H$ be graphs without isolated vertices.
The Ramsey number $R(G,H)$ is the minimum integer $N$ such that  any coloring of the edges of $K_N$ in red or blue yields a red $G$ or a blue $H$.
Determining $R(G,H)$ in general is a very challenging problem, there are several excellent surveys on Ramsey numbers.
In this paper,  we consider a problem related to Ramsey goodness.
In  \cite{B}, Burr gave the following lower bound.
\begin{theo}\label{Burr} {\rm(Burr \cite{B})}
For a connected graph $G$ and a graph $H$ with $v(G)\geq s(H)$,
$$R(G,H) \geq (v(G)-1)(\chi(H)-1)+s(H).$$
\end{theo}

Burr defined $G$ to be $H$-good if the equality $$R(G,H) = (v(G)-1)(\chi(H)-1)+s(H)$$ holds under the conditions of Theorem \ref{Burr}.

In the 1970s, before the definition of Ramsey-goodness  was given, a well-known result of Chv\'atal \cite{C} showed that any tree is $K_m$-good for every $m\geq2$ and an earlier result of Chv\'atal and Harary \cite{CH} showed that any tree is $2K_2$-good.
By applying Chv\'atal's theorem, Stahl \cite{St}  determined the Ramsey number of a forest versus $K_m$.
In 1983,  Burr and Erd\H{o}s \cite{BE} proved that for any fixed $k$ and $m$, there exists $n_0$ such that the family of connected graphs with bandwidth at most $k$ and at least $n_0$ vertices is $K_m$-good, where the bandwidth of a graph $G$ is the smallest number $k$ such that there is an ordering $v_1,\dots,v_n$ of $V(G)$ such that each edge $v_iv_j$ satisfies $|v_i-v_j|\leq k$.
This result was recently extended by Allen, Brightwell and Skokan \cite{ABS} that for each fixed graph $H$ and $k$,  there exists $n_0$ such that the family of connected graphs with bandwidth at most $k$ and at least $n_0$ vertices is $H$-good.
These two results implies a path (with bandwidth  $1$) or a cycle (with bandwidth  $2$) with sufficiently  large number of vertices has good goodness properties.
Results without the assumption of  sufficiently  large number of vertices are also interesting.
In \cite{SAM}, Sudarsana, Adiwijaya and Musdalifah  showed that $P_n$ is $2K_m$-good for $n\geq3$ and $m\geq2$, and conjectured that any tree $T_n$ with $n$ vertices is $2K_m$-good.
Recently, Pokrovskiy and Sudakov \cite{PS} proved that for a fixed graph $H$, the path on $n$ vertices  with $n\geq4v(H)$ is $H$-good.
Balla, Pokrovskiy and  Sudakov \cite{BPS} showed that for all $\Delta$ and $k$, there exists a constant $C_{\Delta,k}$ such that for any tree $T$ with maximum degree at most $\Delta$ and any $H$ with $\chi(H)=k$ satisfying $v(T)\geq C_{\Delta,k}v(H)\log^4 v(H)$, $T$ is $H$-good.
In \cite{LLD}, Lin,   Li and  Dong proved that if $T_n$ is $G$-good and $s(G)=1$, then $T_n$ is $K_1\vee G$-good.
For other results concerning Ramsey-goodness graphs, we refer the reader to the survey papers by Conlon, Fox and Sudakov \cite{CFS}, and Radziszowski \cite{R}.
In this paper, we consider the Ramsey number of a forest versus disjoint union of complete graphs.
In Section $2$, we show  the following result  which is  a continuation of Chv\'atal's  classical result.
It also  confirms the mentioned  conjecture  by Sudarsana, Adiwijaya and Musdalifah \cite{SAM}.

\begin{theo}\label{Tn}
Let $n\geq 3$ and $m\geq2$ be integers. Let $T_n$ be a tree with order $n$, then $$R(T_n,2K_m)=(n-1)(m-1)+2.$$
\end{theo}

The  proof of Theorem \ref{Tn} will be given in Section $2$.
In Section $3$, we show a  result which yields that $T_n$ is $K_m\cup K_l$-good, where $n\geq 3$ and $m>l\geq2$ are integers.
In Section $4$, we introduce a construction which yields a general lower bound of $R(G,H)$ for arbitrary graphs $G$ and $H$.
On  this foundation, we extend the Ramsey goodness of  connected graphs to disconnected graphs and  explore the relation between the Ramsey number of a disconnected graph $\mathrm{F}$ versus a graph $H$ and the Ramsey number of its components versus $H$.
We extend an upper bound given by Gould and Jacobson \cite{GJ},  and show that if each component of a graph $F$ is $H$-good, then $F$ is $H$-good.
Furthermore, we  will apply the Ramsey goodness results of trees to obtain the Ramsey number of a forest versus disjoint union of complete graphs.  Next, we outline the  idea of the proof of Theorem \ref{Tn}.

\subsection{Two Operations on a Tree and Outline of the Proof of Theorem \ref{Tn}}
A  key observation in the proof of Theorem \ref{Tn} is  that any  tree with $n$ vertices can be obtained from $P_n$ by performing a series of two operations.
Furthermore, we show that these two operations preserve the “$2K_m$-goodness” property.
Let us describe these two operations  precisely below.

\

\noindent{\bf Stretching a tree $T$ at a leaf $a$ :} {\em Let $T$ be a tree with $n\geq 3$ vertices and $a$ be a leaf in $T$.
Let $T'$ be obtained from $T$ by deleting a leaf $b$ (other than $a$) of $T$ and adding a new vertex connecting to $a$. We say that $T'$ is obtained by Stretching $T$ at  leaf $a$.}

\

\noindent{\bf Expanding a tree $T$ at a vertex $u$ :} {\em Let $2\leq d\leq n-2$ and $T$ be a tree with $n\geq4$ vertices.  Let $u\in V(T)$ with $N(u)=\{z_0,z_1,\dots,z_{d-1}\}$ satisfying that $z_i$ is a leaf of $T$ for each $i\in[d-1]$ and $z_0$ is not a leaf.
Let $T'$ be obtained from $T$ by deleting a leaf $b\in T- N[u]$ and adding a new vertex connecting to $u$. We say that $T'$ is obtained by Expanding $T$ at vertex $u$.}

\begin{defi}\label{defi}
We say that a property $\mathcal{P}$ is Stretching-preserving (or Expanding-preserving) if a tree $T$ satisfying $\mathcal{P}$ implies that $T'$ also satisfying  $\mathcal{P}$, where $T'$ is obtained by Stretching (or Expanding) $T$ at a leaf (or a vertex).
\end{defi}

 In Section $2.2$, we will prove the following key observation.
\begin{prop}\label{remark}
Given a tree $T$ on $n$ vertices, we can obtain any tree on $n$ vertices from $T$ by applying Stretching and Expanding multiple times.
\end{prop}

It's easy to see that  Proposition \ref{remark} will imply the following corollary.

\begin{coro}\label{111}
If a tree $T$ satisfies property $\mathcal{P}$, and $\mathcal{P}$ is Stretching-preserving  and Expanding-preserving, then every tree satisfies $\mathcal{P}$.
\end{coro}

In Section $2$, we show that the property  “$2K_m$-goodness” is Stretching-preserving  and Expanding-preserving, and prove Theorem \ref{Tn}.

\section{Tree is $2K_m$-good}
\subsection{Proof of Theorem \ref{Tn}}
In this subsection we prove Theorem \ref{Tn}.
We first prove the  crucial lemmas that the property “$2K_m$-goodness” is Stretching-preserving and Expanding-preserving  under some conditions.
\begin{lemma}\label{1}
Let $n\geq 3$ and $m\geq2$ be integers.
Assume that $R(T,2K_{m-1})\leq (n-1)(m-2)+2$ holds for any tree $T$ with order $n$.
Let $T_n^*$ be a tree with $n$ vertices and $T_n^{**}$ be obtained by Stretching $T_n^{*}$ at a leaf $a$  (see Figures \ref{T3} and \ref{T4}).
If $R(T_n^*,2K_m)\leq (n-1)(m-1)+2$, then $R(T_n^{**},2K_m)\leq (n-1)(m-1)+2$.
 \end{lemma}
\begin{proof}
If $n=3$, a tree with $3$ vertices must be $P_3$ and the result holds.
If $T_n^*=P_n$ is a path, then $T_n^{**}=P_n$ and the result holds.
So we may assume that $n\geq4$ and $T_n^*$ is not a path.

Let $u$ be the vertex adjacent to $a$ in $T_n^*$.
Since $T_n^*$ is not a path and $n\geq4$, $T_n^*$ has at least two leaves $b$ and $c$ in $V(T_n^*)\setminus\{a,u\}$.
Let $vc$ be an edge (as in Figure \ref{T3}).
We remark that it's possible that $v=u$.
\begin{figure}[H]
\centering
\begin{minipage}{5cm}
\includegraphics[width=0.9\textwidth, height=0.7\textwidth]{T3}
\caption{$T_n^*$}
   \label{T3}
\end{minipage}
\begin{minipage}{5cm}
\includegraphics[width=0.87\textwidth, height=0.7\textwidth]{T4}
\caption{$T_n^{**}$}
   \label{T4}
\end{minipage}
\end{figure}
\noindent
Let $N=(n-1)(m-1)+2$ and the edges of $K_N$ be colored by red or blue.
We will show that $K_N$ contains a red $T_n^{**}$ or a blue $2K_m$.
Since $R(T_n^*,2K_m)\leq (n-1)(m-1)+2$, then $K_N$ contains a red $T_n^{*}$ or a blue $2K_m$.
We just need to consider that $K_N$ contains a red $T_n^{*}-\{b\}\subseteq T_n^{*}$, whose order is $n-1$.
Since $N-(n-1)=(n-1)(m-2)+2$ and $R(T,2K_{m-1})\leq (n-1)(m-2)+2$ holds for any tree $T$ with order $n$, then $K_N-(T_n^{*}-\{b\})$ contains a red $T_n^{**}$ or a blue $2K_{m-1}$.
We just need to consider that $K_N-(T_n^{*}-\{b\})$ contains a blue $2K_{m-1}$, denoted by $A$ and $B$ (see Figure \ref{T5}).
We note that the edges between $a$ and $A$, and $a$ and $B$ are all blue.
Otherwise there will be a red copy of $T_n^{**}$ and we are done.
Let $F=K_N-(T_n^{*}-\{b\})-A-B+\{c\}$.
Clearly $|V(F)|=(n-3)(m-2)+1$.
By Chv\'atal's theorem, $F$ contains a red $T_{n-2}=T_n^{**}-\{b',c'\}$ or a blue $K_{m-1}$.

Case $1$. $F$ contains a red $T_n^{**}-\{b',c'\}$ as in Figure \ref{T5}.
Note that there exist $x_A'\in A$ and $x_B'\in B$ such that $x_A'v'$ and $x_B'a'$ are red.
Otherwise, $\{v'\}\cup A$ and $\{a\}\cup B$ or $\{a'\}\cup B$ and $\{a\}\cup A$ form a blue copy of $2K_m$, we are done.
Hence $T_n^{**}-\{b',c'\}+\{x_A',x_B'\}$ contains a red copy of $T_n^{**}$ (see Figure \ref{T5}), and we are done.
\begin{figure}[H]
\centering
    \includegraphics[height=5.5cm]{T5}
    \caption{$F$ contains a red $T_n^{**}-\{b',c'\}$}
        \label{T5}
    \end{figure}

Case $2$. $F$ contains a blue $K_{m-1}$, denoted by $C$ (see Figure \ref{T6}).
To avoid a blue copy of $2K_m$, there exist $x_A\in A$ and $x_B\in B$ such that $ux_A$ and $vx_B$ are red.
Otherwise, $\{u\}\cup A$ and $\{a\}\cup B$ or $\{v\}\cup B$ and $\{a\}\cup A$ form a blue copy of $2K_m$, we are done.
Furthermore, there exists $x_C\in C$ such that $x_Ax_C$ is red.
Otherwise, $\{x_A\}\cup C$ and $\{a\}\cup B$ form a blue copy of $2K_m$, we are done.
Hence $T_n^{*}-\{b\}-\{a,c\}+\{x_A,x_B,x_C\}\subseteq K_N$ contains a red copy of $T_n^{**}$ (see Figure \ref{T6}). and we are done.
\begin{figure}[H]
\centering
    \includegraphics[height=5.5cm]{T6}
    \caption{$F$ contains a blue $K_{m-1}$}
        \label{T6}
    \end{figure}
\end{proof}

\begin{lemma}\label{2}
Let $n\geq 3$ and $m\geq2$ be integers.
Assume that $R(T,2K_{m-1})\leq (n-1)(m-2)+2$ holds for any tree $T$ with order $n$.
Let $T_n^*$ be a tree with $n$ vertices.
Let $T_n^{**}$ be obtained by Expanding $T_n^{*}$ at a vertex $u$ (see Figures \ref{T1} and \ref{T2}).
If $R(T_n^*,2K_m)\leq (n-1)(m-1)+2$, then $R(T_n^{**},2K_m)\leq (n-1)(m-1)+2$.
\begin{figure}[H]
\centering
\begin{minipage}{5cm}
\includegraphics[width=0.85\textwidth, height=0.8\textwidth]{T1}
\caption{$T_n^*$}
   \label{T1}
\end{minipage}
\begin{minipage}{5cm}
\includegraphics[width=0.87\textwidth, height=0.8\textwidth]{T2}
\caption{$T_n^{**}$}
   \label{T2}
\end{minipage}
\end{figure}
\end{lemma}
\begin{proof}
We just need to consider that $n\geq4$.
Since $T_n^*- N_1(u)+\{z_0\}$ is also a tree, then there exists at least a leaf $b$ other than $z_0$ in $T_n^*- N_1(u)$.
Note that $b$ is also a leaf of $T_n^*$.
Recall that $d\leq n-2$ in view of the definition of Expanding.
Let $N=(n-1)(m-1)+2$ and the edges of $K_N$ be colored by red or blue.
We will show that $K_N$ contains a red $T_n^{**}$ or a blue $2K_m$.
Since $R(T_n^*,2K_m)\leq (n-1)(m-1)+2$,
then $K_N$ contains a red $T_n^{*}$ or a blue $2K_m$.
We just need to consider that $K_N$ contains a red $T_n^{*}-\{b\}\subseteq T_n^{*}$,
whose order is $n-1$.
Since $N-(n-1)=(n-1)(m-2)+2$ and $R(T,2K_{m-1})\leq (n-1)(m-2)+2$ holds for any tree $T$ with order $n$,
then $K_N-(T_n^{*}-\{b\})$ contains a red $T_n^{**}$ or a blue $2K_{m-1}$.
We just need to consider that $K_N-(T_n^{*}-\{b\})$ contains a blue $2K_{m-1}$, denoted by $A$ and $B$ (see Figure \ref{T7}).
We note that the edges between $u$ and $A$, and $u$ and $B$ are all blue.
Otherwise there will be a red copy of $T_n^{**}$ and we are done.
\begin{figure}[H]
\centering
    \includegraphics[height=4cm]{T7}
    \caption{$K_N-(T_n^{*}-\{b\})$ contains a blue $2K_{m-1}$}
        \label{T7}
    \end{figure}

Let $H_1=K_N-(T_n^{*}-\{b\})-A-B+\{z_1\}$.
Then $|V(H_1)|=(n-3)(m-2)+1$.

If $d\geq3$, we perform the following procedure.
By Chv\'atal's theorem, $H_1$ contains a red $T_{n-2}=T_n^{**}-\{z_1',z_2'\}$ or a blue $K_{m-1}$.
Let us consider that $H_1$ contains a red $T_n^{**}-\{z_1',z_2'\}$ first, as in Figure \ref{T8}.
Note that there exist $x_A'\in A$ and $x_B'\in B$ such that $x_A'u'$ and $x_B'u'$ are red.
Otherwise, $\{u'\}\cup A$ and $\{u\}\cup B$ or $\{u'\}\cup B$ and $\{u\}\cup A$ form a blue copy of $2K_m$, we are done.
Hence $T_n^{**}-\{z_1',z_2'\}+\{x_A',x_B'\}$ contains a red copy of $T_n^{**}$ (see Figure \ref{T8}), and we are done.
\begin{figure}[H]
\centering
    \includegraphics[height=5cm]{T8}
    \caption{$H_1$ contains a red $T_n^{**}-\{z_1',z_2'\}$}
        \label{T8}
    \end{figure}
\noindent Now we consider that $H_1$ contains a blue $K_{m-1}$, denoted by $C_1$ (see Figure \ref{T9}).
Let $H_{2}=H_1-C_1+\{z_2\}$ (see Figure \ref{T9}).
Then $|V(H_2)|=(n-4)(m-2)+1$.
By Chv\'atal's theorem, $H_2$ contains a red $T_{n-3}=T_n^{**}-\{z_1',z_2',z_3'\}$ or a blue $K_{m-1}$.
Let us consider that $H_2$ contains a red $T_n^{**}-\{z_1',z_2',z_3'\}$ first, as in  Figure \ref{T9}.
Note that there exist $x_A'\in A$, $x_B'\in B$ and $x_{C_1}'\in C_1$ such that $x_A'u'$, $x_B'u'$ and $x_{C_1}'u'$ are red.
Otherwise, $\{u'\}\cup A$ and $\{u\}\cup B$ or $\{u'\}\cup B$ and $\{u\}\cup A$ or $\{u'\}\cup C_1$ and $\{u\}\cup A$ form a blue copy of $2K_m$, and we are done.
Hence $T_n^{**}-\{z_1',z_2',z_3'\}+\{x_A',x_B',x_{C_1}'\}$ contains a red copy of $T_n^{**}$ (see Figure \ref{T9}), and we are done.
\begin{figure}[H]
\centering
    \includegraphics[height=5cm]{T9}
    \caption{$H_2$ contains a red $T_n^{**}-\{z_1',z_2',z_3'\}$}
        \label{T9}
    \end{figure}
\noindent  We now consider that $H_2$ contains a blue $K_{m-1}$, denoted by $C_2$.
We take a similar procedure as before and continue the process at most $d-2$ times.
Either we are done or we obtain $d-2$ blue copies of $K_{m-1}$, denoted by $C_1,C_2,\dots,C_{d-2}$ (see Figure \ref{T10}) and $H_1,H_2,\dots,H_{d-2}$ with $H_{i+1}=H_i-C_i+\{z_{i+1}\}$ for $i=1,2,\dots,d-3$.
Let $H_{d-1}=H_{d-2}-C_{d-2}+\{z_{d-1}\}$.
If $d=2$, then $H_1=K_N-(T_n^{*}-\{b\})-A-B+\{z_1\}$, as in the beginning of the last paragraph.
Then $|V(H_{d-1})|=[n-2-(d-1)](m-2)+1=(n-d-1)(m-2)+1$.
Recall that $d\leq n-2$, then $|V(H_{d-1})|\geq m-1$.
By Chv\'atal's theorem, $H_{d-1}$ contains a red $T_{n-d}=T_n^{**}-\{z_1',z_2',\dots,z_d'\}$ or a blue $K_{m-1}$ (see Figure \ref{T10}).

Let us consider that $H_{d-1}$ contains a red $T_n^{**}-\{z_1',z_2',\dots,z_d'\}$ first.
Note that there exist $x_A'\in A,x_B'\in B,x_{C_1}'\in C_1,\dots,x_{C_{d-3}}'\in C_{d-3}$ and $x_{C_{d-2}}'\in C_{d-2}$, such that $x_A'u',x_B'u',x_{C_1}'u',\dots,x_{C_{d-3}}'u'$ and $x_{C_{d-2}}'u'$ are red.
Otherwise,
$\{u'\}\cup A$ and $\{u\}\cup B$ or $\{u'\}\cup B$ and $\{u\}\cup A$ or $\{u'\}\cup C_i$ and $\{u\}\cup A$ form a blue copy of $2K_m$, where $i\in[d-2]$, and we are done.
Hence $T_n^{**}-\{z_1',z_2',\dots,z_d'\}+\{x_A',x_B',x_{C_1}',\dots,x_{C_{d-3}}',x_{C_{d-2}}'\}$ contains a red copy of $T_n^{**}$ (see Figure \ref{T10}), and we are done.
\begin{figure}[H]
\centering
    \includegraphics[height=5cm]{T10}
    \caption{$H_{d-1}$ contains a red $T_n^{**}-\{z_1',z_2',\dots,z_d'\}$}
        \label{T10}
    \end{figure}
We now consider that $H_{d-1}$ contains a blue $K_{m-1}$, denoted by $C_{d-1}$ (see Figure \ref{T11}).
To avoid a blue copy of $2K_m$, there exists $x_{C_1}\in C_1$ such that $z_0x_{C_1}$ is red.
Otherwise,
$\{z_0\}\cup C_1$ and $\{u\}\cup B$ form a blue copy of $2K_m$ and we are done.
Furthermore, there exist $x_A\in A,x_B\in B,x_{C_2}\in C_2,\dots,x_{C_{d-1}}\in C_{d-1}$, such that $x_{C_1}x_A,x_{C_1}x_B,x_{C_1}x_{C_2},\dots,x_{C_1}x_{C_{d-1}}$ are red.
Otherwise,
$\{x_{C_1}\}\cup A$ and $\{u\}\cup B$ or $\{x_{C_1}\}\cup B$ and $\{u\}\cup A$ or $\{x_{C_1}\}\cup C_i$ and $\{u\}\cup A$ form a blue copy of $2K_m$, where $i\in[d-1]\setminus\{1\}$.
Then $T_n^{*}-\{b\}-\{u\}-\{z_1,\dots,z_{d-1}\}+\{x_{C_1}\}+\{x_A,x_B,x_{C_2},\dots,x_{C_{d-1}}\}$ contains a red copy of $T_n^{**}$ (see Figure \ref{T11}), and we are done.
\begin{figure}[H]
\centering
    \includegraphics[height=5cm]{T11}
    \caption{$T_n^{*}-\{b\}-\{u\}-\{z_1,\dots,z_{d-1}\}+\{x_{C_1},x_A,x_B,x_{C_2},\dots,x_{C_{d-1}}\}$ contains a red copy of $T_n^{**}$}
        \label{T11}
    \end{figure}
\end{proof}

Before giving the proof of Theorem \ref{Tn}, we show the following result given by  Sudarsana, Adiwijaya and Musdalifah \cite{SAM}.

\begin{lemma}\label{Pn}
Let $n\geq3$ and $m\geq2$ be positive integers, then $R(P_n,2K_m)=(n-1)(m-1)+2$.
\end{lemma}
\begin{proof}
We just need to prove the upper bound since the lower bound follows  by Theorem  \ref{Burr}  directly.
Let $N=(n-1)(m-1)+2$ and the edges of $K_N$ be colored by red or blue.
We will show that $K_N$ contains a red $P_n$ or a blue $2K_m$.
We first claim that there is a red $P_{n-1}$ contained in $K_N$.
By the result given by Chv\'atal \cite{C}  that $R(T_n,K_m)=(n-1)(m-1)+1$, $K_N$ contains a red $P_n$ or a blue $K_m$.
We just need to consider that $K_N$ contains a blue $K_m$ (denote it by $K$), otherwise we are done.
Since $N-m=(n-2)(m-1)+1$ and we continue to apply the result of Chv\'atal \cite{C}  that $R(P_{n-1},K_m)=(n-2)(m-1)+1$,  then the graph induced on $V(K_N)-V(K)$ contains either a blue $K_m$ and we are done, or a red $P_{n-1}$.
Hence, there is a red $P_{n-1}$ contained in $K_N$ and we denote it  by $P$. Let us label the end vertices of $P$ as $v_1$ and $v_2$.
Now we use induction on $m$ to complete the proof.
By a result of Chv\'atal and Harary \cite{CH},  the conclusion holds for $m=2$.
Assume that the conclusion holds for $m-1$ and we will show that it holds for $m$.
Let $H$ be the graph induced on $V(K_N)-V(P)$, then $v(H)=(n-1)(m-2)+2$.
Since  the conclusion holds for $m-1$, then we obtain either a red $P_{n}$ and we are done, or a blue $2K_{m-1}$ in $H$.
We denote the blue $2K_{m-1}$ by $A$.
Now we consider the edges between $\{v_1,v_2\}$ and $V(A)$.
To avoid a red $P_n$, all edges between $\{v_1,v_2\}$ and $V(A)$ are blue, which will imply a blue $2K_m$ and we are done.
\end{proof}

{\bf Proof of Theorem \ref{Tn}} \ By  Lemma \ref{Pn}, we just need to consider that $n\geq4$ and $T_n$ is not a path.
The lower bound holds by Theorem \ref{Burr}.
Now we use induction on $m$ to prove the upper bound.
By
a result of Chv\'atal and Harary \cite{CH},  the conclusion holds for the case that $m=2$.
Now we assume that the conclusion holds for $m-1$, i.e., $R(T,2K_{m-1})\leq (n-1)(m-2)+2$ holds for any tree $T$ with $n$ vertices.
Under this assumption, Lemma \ref{1} and Lemma \ref{2} guarantee that the property  “$2K_m$-good” is Stretching-preserving  and Expanding-preserving.
Applying Lemma \ref{Pn} that $P_n$ is $2K_m$-good and Corollary \ref{111}, we obtain that $R(T_n,2K_{m})\leq (n-1)(m-1)+2$.
\hskip 1.3em\hfill\rule{6pt}{6pt}

\subsection{Obtain $T_n$ from $P_n$ by Stretching and Expanding}
In this subsection, we give the proof of Proposition \ref{remark}.
\

{\bf Proof of Proposition \ref{remark}} \
For any fixed tree $T_n$ on $n$ vertices, let $P$ be a longest path in $T_n$.
Let us label the end vertices of $P$ as $v$ and $w$, and label the vertices with degree greater than $2$ in the path as $u_1,u_2,\dots,u_l$.
Assume that the distance between $v$ and $u_1$, denoted by $d(v,u_1)$, is $t_0+1$, and $d(u_i,u_{i+1})=t_i+1$ for each $i\in[l-1]$ and $d(u_l,w)=t_l+1$.
Let $d_i=d(u_i)-2$.
We label the vertices not in $P$ and connecting to $u_i$ as $y_1^i,y_2^i,\dots,y_{d_i}^i$.
For each $i\in[l]$, let $B^i$ be a subset of $V(T_n)$ such that the subgraph induced on $\{u_i\}\cup B^i$ is a subtree of $T_n$ and this subtree just contain one vertex of $P$, i.e.,  $u_i$.
Clearly, $V(T_n)=V(P)\cup B^1\cup\dots\cup B^l$ as in Figure \ref{T12}.
\begin{figure}[H]
\centering
    \includegraphics[height=4.5cm]{T12}
     \caption{$T_n$}
     \label{T12}
    \end{figure}
    It's  easy to see that $P_n$ can be obtained from  $T_n$ by applying Stretching multiple times.
     Actually, for any $i\in[l]$, the subgraph induced on $\{u_i\}\cup B^i$ is a subtree of $T_n$, thus there is at least one leaf in every such subtree.
   Firstly, we can Stretching $T_n$ at the leaf $v$ by deleting a leaf $v'\in B^1$, then we lengthen path $P$ with   the end vertices as $v'$ and $w$.
   We continue  Stretching this tree at the leaf $v'$ by deleting a leaf in $B^1-\{v'\}$ (if $B^1-\{v'\}=\emptyset$, then we deleting a leaf in $B^2$).
   Continue this progress, we can   obtain  $P_n$ from  $T_n$ by applying Stretching multiple times (clearly the number of operations is equal to $|B^1|+\dots+|B^l|$).

     Now we prove  that $T_n$ can be obtained from  $P_n$ by applying Stretching and Expanding multiple times.  We label the vertices of $P_n$ as in Figure \ref{T14} (if $t_1=0$, then $x_1^1$ is $u_2$).
     We will delete all vertices in $\{b_{n-t_0-3},\dots,b_1\}$ in turn  and add all corresponding vertices to obtain $T_n$.
     \begin{figure}[H]
\centering
    \includegraphics[height=1.7cm]{T14}
    \caption{$P_n$}
        \label{T14}
    \end{figure}
     Our general operation is divided into the following steps.

     Step $1$. By applying Stretching and Expanding multiple times, add all vertices in $B^1$.
     We first relabel all vertices of $B^1$ in $T_n$. We relabel $y_{i}^1=i$ for all $i\in[d_1]$ and relabel all vertices in $B^1-\{1,2\dots,d_1\}$ as $d_1+1,d_1+2,\dots,|B^1|$ so that every $j$ is connected to exactly one of the vertices $1,2,\dots,j-1$, where $j\in[|B^1|]\setminus[d_1]$.
       We define $$N_k(u_1)=\{j\in B^1:d(j,u_1)=k\}.$$
Denote $|N_k(u_1)|=N_k$ for all $k\geq 1$.
   Clearly, $N_1(u_1)=\{1,2,\dots,d_1\}$, $N_2(u_1)=\{d_1+1,\dots,d_1+N_2\},\dots$.
   We first show that how to delete $d_1$ vertices in $P_n$ and add all corresponding  vertices in $N_1(u_1)$.
  We   apply Expanding $P_n$ at vertex $u_{1}$ $d_1$ times by deleting $b_{n-t_0-3},b_{n-t_0-4},\dots$ and $b_{n-t_0-(d_1+2)}$, and add $1,2,\dots$ and $d_1$ connecting to $u_{1}$ in turn, then we obtain $T'_n$ as in  Figure \ref{T15}.
\begin{figure}[H]
\centering
    \includegraphics[height=3.5cm]{T15}
    \caption{$T'_n$}
        \label{T15}
    \end{figure}
    \noindent       If for each $i\in[d_1]$, $i$ is a leaf of $T_n$, then we are done and go to step $2$.
   Now we assume that there is $i$  not being  a leaf of $T_n$  for some $i\in[d_1]$.
We assume that we have added all vertices in $N_k(u_1)$, and show that how to add all vertices in $N_{k+1}(u_1)$, where $k\geq1$.
For $v\in N_k(u_1)$ so that the degree of $v$ in $T_n$ is more than $1$,
we perform Stretching the tree at every such $v$ (in turn) by deleting an end point $b_\ell\in P_n$, where $\ell\in[n-t_0-d_1-3]$.
On this foundation, for $v\in N_k(u_1)$ so that the degree of $v$ in $T_n$ is more than $2$,
we perform Expanding the tree at every such $v$ $d_{T_n}(v)-2$ times (in turn) by deleting an end point $b_\ell\in P_n$ in turn, where $\ell\in[n-t_0-d_1-3]$.
Hence we add all vertices in $N_{k+1}(u_1)$, where $k\geq1$.

  Step $2$. By applying Stretching multiple times, add all vertices of $\{x^1_2,\dots,x^1_{t_1},u_2,x^2_1\}$.
  We   first apply Stretching the tree obtained  by step $1$ at the end vertex $x^1_i\in  P_n$ (in turn) by deleting $b_{n-t_0-|B^1|-i-2}$, and add $x^1_{i+1}$ connecting to $x^1_i$ in turn, where $i\in[t_1-1]$.
  Next we apply Stretching this tree at the end vertex $x^1_{t_1}$ by deleting $b_{n-t_0-|B^1|-t_1-2}$, and add $u_2$ connecting to $x^1_{t_1}$.
  At last we apply Stretching this tree at the end vertex $u_2$ by deleting $b_{n-t_0-|B^1|-t_1-3}$, and add $x^2_1$ connecting to $u_2$. Hence, we obtain $T''_n$ as in  Figure \ref{T16}.
\begin{figure}[H]
\centering
    \includegraphics[height=4cm]{T16}
    \caption{$T''_n$}
        \label{T16}
    \end{figure}
On this  foundation, we continue to attach $B_i$ to $u_i$ by performing Stretchings and Expandings similar to step $1$ and $2$ for each $i\in\{2,\dots,2l-1\}$  until we obtain $T_n$.

\hskip 1.3em\hfill\rule{6pt}{6pt}

\section{Tree is $K_m\cup K_l$-good}
Firstly, we prove the following theorem which implies that $T_n$ is $K_m\cup K_l$-good for $n\geq 3$ and $m>l\geq2$.

\begin{theo}\label{Tnn}
Let $G$ be a $2K_{m-1}$-good  graph with $n\geq3$ vertices.
If there exist two vertices $u$ and $v$ with degree one in $G$ such that  $G-\{u,v\}$ is $K_m$-good,
then   $G$ is $K_m\cup K_l$-good, where $m>l\geq2$.
\end{theo}

\begin{proof}
By Theorem \ref{Burr}, we just need to prove the upper bound  that $R(G,K_m\cup K_l)\leq (n-1)(m-1)+1$.
Let $N=(n-1)(m-1)+1$ and the edges of $K_N$ be colored by red or blue.
We will show that $K_N$ contains a red $G$ or a blue $K_m\cup K_{l}$.
Since  $R(G,2K_{m-1})=(n-1)(m-2)+2\leq N$, hence $K_N$ contains a red $G$ or a blue $2K_{m-1}$.
We just need to consider that $K_N$ contains a  blue $2K_{m-1}$. Denote the vertex sets of these two disjoint copies of blue $K_{m-1}$ by $A$ and $B$.
Let $F=K_N-A-B$, then  $|V(F)|=(n-3)(m-1)+1$.
Since $R(G-\{u,v\},K_m)=(n-3)(m-1)+1$, then we just need to consider that $F$ contains a red $G-\{u,v\}$. Let $u'$ and $v'$ be  the vertices connecting with $u$ and $v$ in $G$, respectively (It's possible that $v'=u'$).
 If the edges between $u'$ and $A$ or $v'$ and $B$ are all blue, then we have a blue  $K_m\cup K_{m-1}$.
 Otherwise,  there is at least  one red edge  between $u'$ and $A$, and at least one red edge between $v'$ and $B$, so there is a red $G$ and we are done.
\end{proof}

Taking $G$ to be $T_n$ in Theorem \ref{Tnn}, and applying Theorems \ref{Tn} and \ref{Tnn}, we obtain the following result.
\begin{coro}\label{coro}
$T_n$ is $K_m\cup K_l$-good for $n\geq3$ and $m>l\geq2$.
\end{coro}

\section{The Ramsey number of a forest versus a disjoint union of complete graphs}
In \cite{St}, Stahl  applied Chv\'atal's theorem about the exact value of $R(T_n,K_m)$ to determine the Ramsey number of a forest versus $K_m$.
In this section, we also obtain the Ramsey number $R(F,K_m\cup K_l)$ by applying  Theorems \ref{Tn} and Corollary \ref{coro}, where $F$ is a forest and $m,l\geq2$.
Indeed, we prove a general result under some conditions.
Before that, we extend the Ramsey goodness of  connected graphs to all graphs.
Close behind the lower bound for a connected graph $G$ and $H$ with $|V(G)|\geq s(H)$ given by Burr \cite{B}  that $R(G,H)\geq (|V(G)|-1)(\chi(H)-1)+s(G)$,
Gould and Jacobson \cite{GJ} observed a construction which yields a general lower bound of $R(\mathcal{F},H)$ for arbitrary graph $\mathcal{F}$ and $H$.

\begin{theo}{\rm (Gould-Jacobson \cite{GJ})}\label{GJ}
Let $\mathcal{F}$ be a graph, let $F_1,F_2,\cdots, F_{r-1}$ and $F_r$ be the  components (maximal  connected subgraphs) of $\mathcal{F}$.
Let  $I$ be the set of orders of $F_1,F_2,\cdots, F_r$, i.e., for every $i\in I$, there exists at least a graph $F_l$ with $|V(F_l)|=i$, $l\in[r]$.
Let $n(\mathcal{F})$ be the maximum element in $I$ and $k_i(\mathcal{F})$ be the number of components with order $i$.
Let $H$ be a graph, let  $\beta_i=R(F_i,H)-(|V(F_i)-1|)(\chi(H)-1)-s(H)$ and
$$p=\max_{ j\in I}\Bigg\{(j-1)(\chi(H)-2)+\sum_{i=j}^{n(\mathcal{F})}ik_i(\mathcal{F})\Bigg\}+s(H)-1.$$  then $$p\leq R(\mathcal{F},H)\leq p+\max_{i}(\beta_i).$$
\end{theo}

We say that a graph $\mathcal{F}$ is $H$-good if the equality  $$R(\mathcal{F},H)= \max_{ j\in I}\Bigg\{(j-1)(\chi(H)-2)+\sum_{i=j}^{n(\mathcal{F})}ik_i(\mathcal{F})\Bigg\}+s(H)-1$$ holds under the conditions of Theorem \ref{GJ} and $s(H)\geq j_0$, where $j_0\in I$ be the maximum integer satisfying $(j_0-1)(\chi(H)-2)+\sum_{i=j_0}^{n(\mathcal{F})}ik_i(\mathcal{F})=\max_{ j\in I}\left\{(j-1)(\chi(H)-2)+\sum_{i=j}^{n(\mathcal{F})}ik_i(\mathcal{F})\right\}$.
This definition is consistent for a connected graph with the definition of Ramsey-goodness given by Burr.
We extend the upper bound given by Theorem \ref{GJ}.
Before giving the upper bound  of $R(\mathcal{F},H)$ (Theorem \ref{mathcal{F}}),
we remark the following consequence which is implicit in  Theorem \ref{GJ}.
\begin{remark}\label{5.1}
Let  $H$ be a graph and let $F$ be the disjoint union of graphs $F_1, F_2,\dots, F_k$,  where each of  $F_1,F_2,\cdots, F_k$ has $n$ vertices.
Then
$$R(F,H)\leq \max_{i\in[k]}\left\{R(F_i,H)\right\}+n(k-1).$$
 Moreover,  if  $F_i$ is a connected and $H$-good graph with $n\geq s(H)$ for each $i\in[r]$, then  $F$ is $H$-good.
\end{remark}

\begin{theo}\label{mathcal{F}}
Let $H$ be a graph, let $\mathcal{F}$ be the disjoint union of  graphs $F_1,F_2,\cdots, F_{r-1}$ and $F_r$.
Let  $I$ be the set of orders of $F_1,F_2,\cdots, F_{r}$, i.e., for every $i\in I$, there exists at least a graph $F_l$ with $|V(F_l)|=i$, $l\in[r]$.
Let $n(\mathcal{F})$ be the maximum element in $I$ and $k_i(\mathcal{F})$ be the number of graphs with order $i$. Then
$$R(\mathcal{F},H)\leq\max_{ j\in I}\Bigg\{\max_{|V(F_p)|=j,p\in[r]}\left\{R(F_p,H)\right\}+\sum_{i=j}^{n(\mathcal{F})}ik_i(\mathcal{F})-j\Bigg\}.$$
Moreover, if  $F_i$ is a connected and $H$-good graph with $|V(F_i)|\geq s(H)$ for each $i\in[r]$, then  $\mathcal{F}$ is $H$-good.
\end{theo}

\begin{proof}
 Let  $$N=\max_{ j\in I}\Bigg\{\max_{|V(F_p)|=j,p\in[r]}\left\{R(F_p,H)\right\}+\sum_{i=j}^{n(\mathcal{F})}ik_i(\mathcal{F})-j\Bigg\}$$ and the edges of $K_N$ be colored by red or blue.
Now we  prove the upper bound by assuming that there is no blue $H$ and using the descending  induction to show the existence of a red $\mathcal{F}$ in $K_N$.
Let $G_j$ be the  graph consisting of all $F_i$ with order at least $j$.
Clearly,
\begin{eqnarray*}
N&\geq& \max_{|V(F_i)|=n(\mathcal{F})}\left\{R(F_i,H)\right\}+n(\mathcal{F})k_{n(\mathcal{F})}(\mathcal{F})-n(\mathcal{F})\\
&=& \max_{|V(F_i)|=n(\mathcal{F})}\left\{R(F_i,H)\right\}+n(\mathcal{F})(k_{n(\mathcal{F})}(\mathcal{F})-1)\\
& \overset{Remark\; \ref{5.1}}{\geq}& R(G_{n(\mathcal{F})},H).
\end{eqnarray*}
Apply the assumption that there is no blue $H$, there is a red $G_{n(\mathcal{F})}$ in $K_N$.
So the base case holds.
Now  assume that there is a red $G_{j+1}$ in $K_N$, our goal is to show that there is a red $G_{j}$ in $K_N$.
Since $|V(G_{j+1})|=\sum_{i=j+1}^{n(\mathcal{F})}ik_i(\mathcal{F})$, then the order of $K_N-G_{j+1}$ is
\begin{eqnarray*}
N-\sum_{i=j+1}^{n(\mathcal{F})}ik_i(\mathcal{F})&\geq& \max_{|V(F_p)|=j,p\in[r]}\left\{R(F_p,H)\right\}+\sum_{i=j}^{n(\mathcal{F})}ik_i(\mathcal{F})-j-\sum_{i=j+1}^{n(\mathcal{F})}ik_i(\mathcal{F})\\
&=& \max_{|V(F_p)|=j,p\in[r]}\left\{R(F_p,H)\right\}+jk_j(\mathcal{F})-j\\
&=& \max_{|V(F_p)|=j,p\in[r]}\left\{R(F_p,H)\right\}+j(k_j(\mathcal{F})-1)\\
& \overset{Remark\; \ref{5.1}}{\geq}& R(G_j-G_{j+1},H).
\end{eqnarray*}
Hence there is a red $G_j-G_{j+1}$ in $K_N-G_{j+1}$, which yields a red $G_j$ in $K_N$.
By induction  the proof of the first part of the theorem  is completed.

If   $F_i$ is a connected and $H$-good graph with $|V(F_i)|\geq s(H)$ for each $i\in[r]$,
then $R(\mathcal{F},H)\leq\max_{ j\in I}\left\{(j-1)(\chi(H)-1)+s(H)+\sum_{i=j}^{n(\mathcal{F})}ik_i(\mathcal{F})-j\right\}.$
Combining with the lower bound of Theorem \ref{GJ}, we obtain that $\mathcal{F}$ is $H$-good.
\end{proof}

By Corollary \ref{coro} and Theorem \ref{mathcal{F}}, we can  obtain the following exact value of $R(F,K_m\cup K_l)$, where $F$ is a forest and $m,l\geq2$, and a forest is $K_m\cup K_l$-good.

\begin{coro}
Let $m\geq l \geq 2$ be positive integers.
Let $F$ be a forest, then $F$ is $K_m\cup K_l$-good.
\end{coro}

\section{Remarks}
Determining $R(G,H)$ in general is a very challenging problem.
When we focus on the  problem related to Ramsey goodness, we are interested in exploring what kind of conditions can yield good Ramsey goodness property of graphs.
As mentioned in Section $1$, nice work in \cite{BPS, C, CH, PS} concern this type of conditions.
Theorem \ref{mathcal{F}} shows that if every component of a disconnected graph $F$ is $H$-good, then $F$ is $H$-good.
It is natural to study whether $F$ is $H$-good and  $G$-good implies that $F$ is $H\cup G$-good.
Clearly, this is not true for graphs such as $K_n$.
An obvious result is that any connected graph $F$ is $K_2$-good,
but $R(K_n,2K_2)=n+2$, which implies that $K_n$ is not $2K_2$-good.
Certainly, some results  provide evidence for that the answer of this question is yes for some  $F$.
Sudarsana \cite{Su} showed that $P_n$ with $n\geq (t-2)((tm-2)(m-1)+1)+3$ is $tK_m$-good for  $m$ and $t\geq2$ be integers.
Indeed, the condition on the number of vertices $n$ can not  be released completely.
For example,  we can prove that $T_n$ is not $tK_2$-good when $n\leq t$ (actually we determined the exact Ramsey number $R(T_n,tK_2)$ in \cite{HP}).
It's interesting to explore what kind of graph $F$ can satisfy that $F$ is $H$-good and $G$-good implies that $F$ is $H\cup G$-good when the number of vertices in $F$ is sufficiently large.


\section{Introduction}

The Very Large Array (VLA) has played a key role in exploring the
radio Universe for almost three decades. To continue the unparalleled
success of this facility, the VLA is currently undergoing a drastic
rejuvenation process to become the 'Expanded VLA' or EVLA. The
conversion is very comprehensive and comprises additional and upgraded
receivers, new broad-band fiber optics, new online control systems,
new digital electronics, and a state-of-the-art correlator called
WIDAR (Wideband Interferometric Digital Architecture). The upgrades
will not only improve the continuum sensitivity by about an order of
magnitude largely due to a substantial increase of instantaneous
receiver and correlator bandwidth, but they will also provide extreme
spectral resolution and a wide coverage of the radio spectrum -- for
the first time it will be possible to observe at any chosen frequency
in the entire 1-50\,GHz radio window.

The EVLA specifications and its current status is described on the following webpage:\\
\verb=http://www.aoc.nrao.edu/evla=\\
(see also \cite{nap06}). With its unparalleled sensitivity the EVLA is
a true pathfinder for the Square Kilometer Array (SKA). Even the
EVLA's mapping speed matches that of special-purpose, ultra-wide field
but lower sensitivity and narrow band SKA pathfinders like ASKAP, ATA,
or MEERKAT in their early incarnations.

One of the fields of astronomy that the VLA pioneered is the
observation of atomic, neutral hydrogen (H{\sc i}) in our Milky Way, in
nearby galaxies and the intragroup medium, as well as at higher
redshifts. Those observations revealed the complex properties of the
neutral ISM, its importance on star formation and the physics of
the interface between a galaxy's disk, its halo and the intergalactic
medium. The abundance of H{\sc i}\ and its 21\,cm line properties also
proved to be indispensable tools to derive the complex dynamics of
galaxies in the form of, for example, density waves, rotation curves
(and thus the dark matter distribution), tidal interactions, and
mergers of galaxies. In this article we like to summarize what EVLA
offers for future H{\sc i}\ observations.

\section{EVLA Receivers for H{\sc i}\ Observations}

The H{\sc i}\ hyperfine line rest frequency of $\sim 1.420$\,GHz falls into
the radio L--band. The VLA L--band receivers cover a frequency range
of 1.25--1.8\,GHz. The EVLA receivers will widen this to
0.93--2.1\,GHz. At the lower end this will increase the H{\sc i}\ redshift
coverage from a current upper limit of $z\sim 0.14$ to $z\sim
0.53$. This is an improvement of almost $\Delta z\sim 0.4$ or an
additional $\sim 3.5$\,Gyr of look-back time (in a WMAP $\Lambda$CDM
cosmology). The system temperature over telescope efficiency T$_{\rm
  sys}/\epsilon$ of the EVLA is expected to decrease from $\sim 75$\,K
to about 60\,K or less (T$_{\rm sys}$ of the EVLA is designed to hover
around $\sim 26$\,K), saving about 30\% of integration time to reach
the same sensitivity.

The old VLA L--band feedhorn design featured a microwave lens in the
optical path. With the redesigned EVLA receivers, this lens is not
required anymore. Thus, ground radiation scattered by the lens (and
other structural dish elements) into the feedhorn is largely reduced
and the EVLA L--band system temperature improves substantially over
the VLA at lower elevations; at an elevation of $\sim 20^{\circ}$ the
EVLA system temperature is about half that of the VLA which
corresponds to a four fold increase in sensitivity.

As part of the EVLA conversion, the L--band receivers will be equipped
with new orthomode transducers. Until they become available for all
antennas in 2012, an interim L--band system is currently being
installed (with a slightly lower sensitivity and a minimum frequency
of 1\,GHz, corresponding to $z\sim0.42$). For more information on the
performance of the EVLA L--band system upgrade and performance, we
would like to refer to the EVLA webpages and also to Emmanuel
Momjian's contribution in this volume.

At lower frequencies the EVLA offers a P--band receiver which covers
frequencies of 300--340\,MHz equaling an H{\sc i}\ $z\sim 3.2-3.7$
redshift range. This band remains unchanged in the VLA to EVLA
conversion. The system is not sensitive enough to observe typical, gas
rich galaxies at the available redshift in H{\sc i}\ emission. However,
searches for H{\sc i}\ absorption in P--band have been conducted in the
past (e.g.\cite{tar94}) and are still an option for the EVLA.

\section{The WIDAR correlator}

Since the installation of the VLA correlator, Moore's law pushed
processing speeds of computers by $\sim 5$ orders of magnitude. Taking
advantage of this, a new correlator, WIDAR, will be commissioned in
2008/2009. However, one should keep in mind that the current VLA correlator
was designed to be very suitable for H{\sc i}\ observations toward nearby
galaxies. The bandwidth and resolution almost perfectly matches what
is needed for such observations (bandwidth of $\sim 200$\,km\,s$^{-1}$\ at a
resolution of $\sim 5$\,km\,s$^{-1}$). But the VLA correlator design severely
limits the amount of 'discovery space', e.g., very wide and shallow
lines (for an example, see \cite{mor07}) would not be discovered with
the VLA without prior knowledge of these features. The narrow
bandwidth typically used for galaxies of $\sim 1.5$\,MHz has only few,
supposedly line--free channels at the band edges. Any wide lines
extending across these channels would not be discovered because they
would be removed in the process of continuum subtraction. Another case
of limited discovery space is that other H{\sc i}\ sources in the field,
e.g., companion galaxies, remain undetected with the VLA if they are
at a slightly different velocity than the main target. At the other
extreme, very narrow line features, e.g., caused by H{\sc i}\
self--absorption are smeared out and would be missed in a typical
extragalactic H{\sc i}\ setup with its velocity resolution of a few
km\,s$^{-1}$. To open up new discovery space, wide bands at high spectral
resolution are desired. Such capabilities are provided by the new
WIDAR correlator (for a description of the technology, see
\cite{car00}). WIDAR will have a spectral resolution of down to Hz
ranges and a bandwidth of up to 8\,GHz. The full bandwidth is split up
into four 2\,GHz baseband pairs and in each baseband pair up to 16
independent sub-band pairs can be selected with bandwidths between
31.25\,kHz and 128\,MHz. The full 8\,GHz baseband pairs have a {\it
  minimum} of 16384 channels which will always be
available. Recirculation trades bandwidth for more channels and the
maximum number of channels are of order 4 million. Such a flexible
design will cover virtually any need for setups to observe H{\sc i}\ in
single targets with thousands of km\,s$^{-1}$\ bandwidth and sub-km\,s$^{-1}$\
velocity resolution.

But the EVLA will be able to do more. The L--band receivers and WIDAR
will cover the entire H{\sc i}\ redshift range of $z=0-0.53$ at a
resolution of 3.2\,km\,s$^{-1}$\ when observed with two polarization products,
and at 6.4\,km\,s$^{-1}$\ when observed at full stokes. Other configurations
will be able to, e.g., stack multiple radio recombination lines in
order to improve the signal--to--noise of Zeeman splitting
experiments in a single observation.

\section{Piggy-backing}

The velocity resolution of a few km\,s$^{-1}$\ over the full 0.93-2.1\,GHz
L--band range enables new, unique synergies with other
observations. Every EVLA L--band continuum observation will also be an
H{\sc i}\ redshift survey and vice versa. Also, virtually every targeted
H{\sc i}\ observation leaves enough computing power in WIDAR to once more
perform a simultaneous H{\sc i}\ survey over the entire $z=0-0.53$ range at
good velocity resolution. As if this would not be enough, the minimum
data dumping time for 1\,GHz bandwidth L--band data is $\sim 100$\,ms
which allows monitoring of and searches for transient sources in the
field while simultaneously observing H{\sc i}\ or radio continuum projects
(but note that current data output limitations imposed by archiving
are $\sim 25$\,MB\,s$^{-1}$, equaling to dumping times of or $\sim
20$\,s, when the maximum number of channels and baselines is read
out). These are exciting new opportunities that will pave the way
toward SKA H{\sc i}\ surveys. For example, without spending any dedicated
survey time, the $z=0-0.53$ redshift volume around a VLA standard
calibrator will accumulate hundreds of hours prior to the
commissioning of the SKA. This will provide very deep H{\sc i}\ and radio
continuum images essentially for free. The up to 8 bit quantization of
WIDAR also delivers improved high dynamic range imaging capabilities
which reduce current sensitivity limitations due to the inevitable
presence of strong sources in any field. The new digital transmission
system also removes the infamous ``3 MHz ripple'' and related spectral
baseline instabilities. This reduces the systematic uncertainties of
deep H{\sc i}\ observations dramatically.

\section{Summary}

Over the VLA, the EVLA will improve H{\sc i}\ observations in terms of a
wider L--band frequency range (down to $\sim 930$\,MHz), a better
$T_{\rm sys}/\epsilon$ sensitivity (in particular at off-zenith
elevations), much improved spectral baseline stability, and, most
importantly, spectral bandwidth and resolution. The L--band receiver
improvement guarantee that the EVLA will still remain the most
sensitive interferometer for H{\sc i}\ in the world for at least a decade,
until SKA pathfinders will be expanded far beyond the currently
planned prototypes. The new WIDAR correlator is flexible enough to
allow observations of virtually any galaxy at sub--km\,s$^{-1}$\ resolution
with thousands of km\,s$^{-1}$\ bandwidth. At the same time, every L--band
continuum or targeted H{\sc i}\ observation will also yield a blind
$z=0-0.53$ redshift H{\sc i}\ survey at a velocity resolution of a few
km\,s$^{-1}$, and vice versa. It is clear that these new possibilities have
their price in a very large data rate and that data reduction will
challenge today's computing capabilities. The opportunities, however,
are tremendous and it is up to the community to develop new strategies
on how to take advantage of the wealth of EVLA data in order to answer
the open questions of galaxy evolution and cosmology.

\begin{theacknowledgments}
The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
\end{theacknowledgments}

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