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by an _ octant _ , in this paper we mean an open subset of @xmath1 of the form @xmath4 and the point @xmath5 is called the _ apex _ of the octant . in @xcite we have shown that every @xmath6-fold covering of a set in @xmath1 by a finite number of octants decomposes into two coverings , i.e. , if every point of some set @xmath7 is contained in at least @xmath6 members of a finite family of octants @xmath8 , then we can partition this family into two subfamilies , @xmath9 , such that every point of @xmath7 is contained in an octant from @xmath10 and in an octant from @xmath11 . we improve this constant in the following theorem , proved in section  [ sec : upper ] . [ thm:9 ] every @xmath0-fold covering of a point set in @xmath1 by finitely many octants decomposes into two coverings . the improvement comes from a careful modification of our original proof for @xmath6-fold covering , while the framework remains essentially the same . the equivalent dual version ( see @xcite ) of this statement is that any finite set of points in @xmath1 can be colored with two colors such that any octant containing at least @xmath0 points contains both colors . it was discovered in a series of papers by cardinal et al . @xcite and by us @xcite that this bound implies several further results for which earlier only doubly exponential bounds were known @xcite . we denote by @xmath12 the smallest integer such that every @xmath12-fold covering of a finite point set in @xmath1 by octants decomposes into two coverings , thus theorem  [ thm:9 ] states that @xmath13 . using this new bound , the degrees of the polynomials in the following theorems have also been improved . ( a diagram describing the connection between different coloring problems can be found later in figure  [ fig : diagram ] . )    for any positive integer @xmath14 and any given triangle @xmath15 , any finite set of points can be colored with @xmath14 colors such that any homothet of @xmath15 containing at least @xmath16 , thus @xmath17 points , contains all @xmath14 colors .    for any positive integer @xmath14 , any @xmath18-fold covering of a subset of @xmath19 by finitely many octants can be decomposed into @xmath14 coverings . this theorem also has the following straight - forward corollaries . * for any positive integer @xmath14 , any finite set of points in @xmath1 can be colored with @xmath14 colors so that any octant with @xmath20 points contains all @xmath14 colors . * for any positive integer @xmath14 , any @xmath20-fold covering of a finite point set in @xmath3 by homothets of a triangle decomposes into @xmath14 coverings . * for any positive integer @xmath14 , any @xmath20-fold covering of a finite point set in @xmath3 by bottomless rectangles decomposes into @xmath14 coverings .    here bottomless rectangle _ refers to a subset of the plane of the form @xmath21 . note that it has been proved by asinowski et al . @xcite that for any positive integer @xmath14 , any finite set of points in @xmath3 can be colored with @xmath14 colors such that any bottomless rectangle containing at least @xmath22 points contains all @xmath14 colors ( they also proved the lower bound that @xmath22 can not be changed to @xmath23 in this statement ) . a very general conjecture ( * ? ? ? * problem 6.7 and after ) implies that all the above parameters can also be replaced by @xmath24 . we also give the following construction , which will be presented in section  [ sec : lower ] . [ thm : lower ] for every triangle @xmath15 there is a finite point set @xmath7 such that for every two - coloring of @xmath7 there is a translate of @xmath15 that contains exactly @xmath2 points and all of these have the same color . this also implies @xmath25 , as the intersection of octants with the plane @xmath26 give all homothets of the triangle @xmath27 @xmath28 @xmath29 , thus if we place the finite point set @xmath7 on this plane , then for any two - coloring of @xmath7 there will be an octant with exactly @xmath2 points , all of the same color . we end the paper by discussing problems about coloring ordered intervals that turn out to be equivalent to the problem of decomposing octants , in section  [ sec : int ] . the dual version of theorem  [ thm:9 ] is that any finite set of points can be colored with two colors such that any octant containing at least @xmath0 points , contains both colors . this is equivalent to the original problem . we will prove the dynamic planar version of the dual problem , which is the following . a _ quadrant _ or _ wedge _ is a subset of the plane of the form @xmath30 . we have to two - color a finite ordered planar point set @xmath31 such that for every @xmath32 every quadrant that contains at least @xmath0 points from @xmath33 contains both colors . this dynamic planar version is also equivalent to the original problem , for the details , see @xcite . ( briefly , the equivalence of the two problems is implied by the following containment - reversing bijections : an octant with apex @xmath5 is bijected to the point @xmath34 that `` appears '' at time @xmath35 , while a point with coordinates @xmath36 is bijected to a mirrored quadrant with apex @xmath36 , i.e. , to the subset @xmath37 . )    a way to imagine this problem is that the points `` appear '' in order and at step @xmath32 we have to color the new point , @xmath38 . this is impossible to do in an online setting @xcite , i.e. , without knowing in advance which points will come in which order . moreover , it was shown by cardinal et al . @xcite that such a coloring is even impossible in a so - called _ semi - online _ model , where points can be colored at any time after their arrival as long as every octant with @xmath0 ( or any other constant number of ) points contains both colors . our strategy , developed in @xcite , builds a forest on the points such that any time any quadrant containing at least @xmath0 points contains two adjacent points from the same tree - component . therefore , after all the points arrived , any proper two - coloring of the forest will be such that any octant containing at least @xmath0 points contains both colors . : the possible relations of two points . ]    : @xmath39 is below the staircase , @xmath40 is above the staircase , @xmath41 and @xmath42 are neighboring staircase points and @xmath41 is the left neighbor of @xmath42 . ] we start by introducing some notation ( see also figure [ fig : comparable ] ) . [ def : comparable ] we say that point @xmath43 is _ northwest _ ( in short nw ) from point @xmath44 if and only if @xmath45 and @xmath46 . in this case we also say that @xmath40 is _ southeast _ ( in short se ) from @xmath39 and that @xmath39 and @xmath40 are _ similarly , we say that point @xmath43 is _ southwest _ ( in short sw ) from point @xmath44 if and only if @xmath45 and @xmath47 . in this case we also say that @xmath40 is _ northeast _ ( in short ne ) from @xmath39 and that @xmath39 and @xmath40 are _ we can suppose that all points have different coordinates , as by a slight perturbation we can only get more subsets of the points contained in a quadrant ( without losing others ) .    at any step @xmath32 , we define a graph @xmath48 ( which is actually a forest ) on the points of @xmath49 and a vertex set @xmath50 of pairwise incomparable points called the _ staircase _ , recursively . at the beginning @xmath51 is the empty graph and @xmath52 is the empty set . a point on the staircase is called a _ stair - point_. thus , before the @xmath53 step we have a graph @xmath54 on the points of @xmath55 and a set @xmath56 of pairwise incomparable points . in the @xmath53 step we add @xmath38 to our point set obtaining @xmath49 and we will define the new staircase , @xmath50 , and also the new graph , @xmath48 , containing @xmath54 as a subgraph . before the exact definition of @xmath50 and @xmath48 , we make some more definitions and fix some properties that will be maintained during the process ( see figure [ fig : stairdef ] ) . [ def : stair ] we say that a point @xmath39 of @xmath49 is _ above _ the staircase if there exists a stair - point @xmath57 such that @xmath39 is ne from @xmath42 . if @xmath39 is not above or on the staircase , then we say that @xmath39 is _ below _ the staircase . a point below ( resp . above ) the staircase is called a _ below - point _ ( resp .  _ above - point _ ) . at any time @xmath32 , we say that two points of @xmath50 are _ neighbors _ if their @xmath58-coordinates are consecutive among the @xmath58-coordinates of the stair - points . ( note that this does not mean that they are connected in the graph . ) we also say that @xmath39 is the _ left _ _ right _ ) neighbor of @xmath40 if @xmath39 and @xmath40 are neighbors and the @xmath58-coordinate of @xmath39 is less ( resp . more ) than the @xmath58-coordinate of @xmath40 .     is good ( resp . right - good , left - good ) if @xmath59 contains two points that are connected by an edge . ]    in any step @xmath32 , we say that a point @xmath39 is _ good _ if any wedge containing @xmath39 already contains two points connected by an edge , which are thus forced to get different colors ( see figure  [ fig : goods ] ) . i.e. , at any time after @xmath32 , a wedge containing @xmath39 will contain points of both colors in the final coloring . a stair - point @xmath39 is _ almost - good _ if for at least one of its neighbors , @xmath40 , it is true that any wedge containing @xmath39 and @xmath40 contains two points connected by an edge of @xmath48 . additionally , if @xmath40 is the left neighbor of @xmath39 , then we say that @xmath39 is _ left - good _ , and if @xmath40 is the right neighbor of @xmath39 , then we say that @xmath39 is _ right - good_.    notice that the good points and the neighbors of the good points are always almost - good . in fact , good points are also left- and right - good , and a left ( resp . right ) neighbor of a good point is right ( resp .  left ) good . now we can state the properties we maintain at any time @xmath32 .    1 . all above - points are good . 2 .   all stair - points are almost - good . 3 .   all below - points are in different components of @xmath48 . @xmath48 is a forest .    for @xmath60 , all these properties are trivially true . whenever a new point arrives , we execute the following operations ( see also figure  [ fig : operations ] ) repeatedly as long as it is possible , in any order . this will ensure that the properties remain true . 1above : : :    if an above - point @xmath39 is not good , then we connect    @xmath39 by an edge with a stair - point that is sw from    @xmath39 . 2comparable : : :    if for some below - points @xmath61 we have that    @xmath40 is ne from @xmath39 , then connect them by an    edge and put @xmath40 on the staircase . 4incomparable : : :    suppose there are no comparable below - points and there is a wedge    @xmath59 that lies entirely below the staircase and contains    four incomparable points , @xmath62 and    @xmath63 , in order of their @xmath58-coordinates . then    connect @xmath64 with @xmath65 and also    @xmath66 with @xmath63 , and put @xmath65    and @xmath66 on the staircase . 1box : : :    suppose there are no comparable below - points , and suppose    @xmath67 and @xmath68 are two neighboring    stair - points , @xmath67 is nw from @xmath68 ,    @xmath67 is left - good but not right - good while    @xmath68 is right - good but not left - good and @xmath39    is a point in the rectangle defined by the two opposite vertices    @xmath67 and @xmath68 . we connect @xmath39    and @xmath68 , and put @xmath39 on the staircase . now we have to verify that the properties remain true after executing an operation . note that this was implicitly proved already in @xcite for _ 1above _ , _ 2comparable _ and _ 4incomparable_. first , we make the following observation , which implies that we will have to verify property 2 only for the new stair - points . [ remainsgood ] if a stair - point is left - good ( resp . right - good , resp . good ) , then if after an operation this point is still a stair - point , then it remains left good ( resp .  right - good , resp . .    notice that if a stair - point gets a new left - neighbor , then the new neighbor is either good , or right - good . similarly , if a stair - point gets a new right - neighbor , then the new neighbor is either good , or left - good . [ szorszalhasogatas ] after doing an operation properties 1 - 4 remain true . we check each operation and each property .    * _ 1above _ : for property 1 , notice that @xmath39 necessarily has to be the newly arrived point , @xmath38 , and it becomes good after the operation . property 2 obviously remains true . for properties 3 and 4 we use that as @xmath39 is the newly arrived point , before the operation it is not connected to any other point . * _ 2comparable _ : property 1 remains true as only points ne from @xmath40 became above - points and thus they all have the edge @xmath69 sw from them . property 2 remains true as the only new stair - point is @xmath40 , which became good . properties 3 and 4 remain true as before the operation @xmath39 and @xmath40 were in different tree - components , which are then connected . * _ 4incomparable _ : property 1 remains true as , using that no below - points were comparable , any point that became an above - point has either both @xmath64 and @xmath65 , or both @xmath66 and @xmath63 sw from it . property 2 remains true as there are only two new stair - points : @xmath65 becomes left - good and @xmath66 becomes right - good . properties 3 and 4 remain true as before the operation @xmath70 were all in different tree - components , and after the operation two - two of these are connected in a suitable way . * _ 1box _ : property 1 remains true as , using that no below - points were comparable , there are no new above - points . property 2 remains true as @xmath39 is the only new stair - point and it is right - good . properties 3 and 4 remain true if @xmath39 and @xmath68 are in different tree - components . this will be proved in lemma  [ remainslost ] and lemma  [ remainslower ] . [ remainslost ] if there is no below - point in the tree - component @xmath71 of a stair - point @xmath42 , then this remains true , i.e. , later during the process the component containing @xmath42 will never contain a below - point . suppose that there is no below - point in the tree - component @xmath71 of a stair - point @xmath42 . this trivially remains true when a new point arrives ( before doing operations ) . then a simple case analysis shows that none of the operations can introduce a below - point to the tree - component @xmath71 of a stair - point @xmath42 :    * _ 1above _ : either @xmath71 does not change or only @xmath39 ( an above - point ) is added to it . * _ 2comparable _ : only the components of the below - points @xmath39 and @xmath40 are joined , as @xmath71 must be a different tree from these two ( as it contained no below - point ) , @xmath71 does not change . * _ 4incomparable _ : only the components of the below - points @xmath72 change , as @xmath71 must be a different tree from these ( as it contained no below - point ) , @xmath71 does not change . * _ 1box _ : either @xmath71 does not change or @xmath73 in which case @xmath71 is joined with the tree @xmath74 containing @xmath39 . in the latter case in @xmath74 the only below - point was @xmath39 ( by property 3 ) , which after the operation becomes a stair - point , so the new tree containing @xmath42 , @xmath75 still does not contain a below - point after this operation . [ remainslower ] suppose @xmath42 is a stair - point and @xmath76 is a below - point in the tree - component @xmath71 containing @xmath42 . if @xmath42 is right - good but not left - good , then @xmath76 is lower than @xmath42 , that is , @xmath76 has a smaller @xmath77-coordinate than @xmath42 .    by observation  [ remainsgood ] , we know that if @xmath42 is right - good but not left - good , then this was also true at the time when @xmath42 became a stair - point . a simple case analysis of the operations shows that at the time when @xmath42 becomes a stair - point , the statement holds :    * _ 1above _ : this is not a possible case as in this case no point becomes a stair - point . * _ 2comparable _ : this is not a possible case as necessarily @xmath42 plays the role of @xmath40 in the operation , in which case @xmath42 is good and thus also left - good , contradicting our assumption on @xmath42 . * _ 4incomparable _ : necessarily @xmath42 plays the role of @xmath66 in the operation , thus @xmath76 is @xmath63 and so it is lower than @xmath42 , as required . * _ 1box _ : necessarily @xmath42 plays the role of @xmath39 in the operation , thus after the operation the below - point in @xmath71 is necessarily the point which was the below point of @xmath78 before the operation . as @xmath68 was right - good but not left - good , by induction this below point was lower than @xmath68 , thus it is also lower than @xmath79 , as required .    if after some step @xmath71 stops to have a below - point , then by lemma  [ remainslost ] this remains true and so there can be no below - point @xmath76 in @xmath71 as required by the lemma and we are done . otherwise , if @xmath71 still has a below - point , then by property 3 there is exactly one below - point @xmath76 in @xmath71 , it is lower than @xmath42 , and we have to check that after any operation the below - point in @xmath71 remains below @xmath42 . the only operation in which the below - point @xmath76 in @xmath71 could go higher is _ 4incomparable _ such that @xmath76 plays the role of @xmath65 . if @xmath80 is sw from @xmath42 , then @xmath42 goes above the staircase , thus stops being a stair - point as required by the lemma and we are done . if @xmath80 is se from @xmath42 , then the whole wedge @xmath59 must be lower than @xmath42 , and then the new below - point in @xmath71 becomes @xmath64 , also lower than @xmath42 . this finishes the proof of the lemma and also of proposition  [ szorszalhasogatas ] . now we can finish the proof of the dynamic dual version , and thus also of theorem  [ thm:9 ] , by showing that taking any ( partial ) two - coloring of the forest @xmath48 constructed using the above operations , at all times ( i.e. , for every prefix set @xmath81 of the point set ) , any quadrant @xmath59 containing at least @xmath0 points contains both colors . fix the time after the arrival of the point @xmath38 ( and after we repeatedly applied the operations as long as possible ) . thus no more operations can be applied , in particular there are no two comparable below - points otherwise we could apply operation _ 2comparable_. if @xmath59 contains an above - point , it contains both colors as all above - points are good . if @xmath59 contains at most one stair - point , @xmath42 , then by `` splitting '' @xmath59 at @xmath42 ( see figure  [ split1 ] ) , we get two quadrants that do not contain any stair - point , but contain all other points that @xmath59 contains . one of these two quadrants must contain at least @xmath2 below - points , thus we could apply operation _ 4incomparable _ , a contradiction . if @xmath59 contains at least @xmath82 stair - points , then it contains a stair - point @xmath42 such that both neighbors of @xmath42 are also in @xmath59 . as every stair - point is almost - good , @xmath59 must contain both colors . finally , if @xmath59 contains exactly two ( neighboring ) stair - points , @xmath67 nw from @xmath68 , then the only way for @xmath59 to be monochromatic is if @xmath67 is left - good but not right - good and @xmath68 is right - good but not left - good . therefore , there can be no points in the rectangle formed by @xmath67 and @xmath68 , as otherwise we could apply operation _ 1box _ , a contradiction . at least one of the two quadrants obtained by `` splitting '' @xmath59 at @xmath67 and @xmath68 ( see figure  [ split2 ] ) , must contain at least @xmath2 below - points , thus we could apply operation _ 4incomparable _ , a contradiction . here we construct for any triangle @xmath15 a finite point set @xmath7 such that for every two - coloring of @xmath7 there is a translate of @xmath15 that contains exactly @xmath2 points and all of these have the same color . as the construction is quite hard to describe precisely , we refer to figure  [ fig : haromszogre4nemeleg ] and figure  [ fig : haromszogre4nemelegzoom ] for the details and give only a more informal description below . with a simple case analysis , we will show that in any two - coloring , there is a monochromatic triangle with exactly @xmath2 points . points.,scaledwidth=100.0% ]        on the top part of the figure is the `` big picture '' that shows what the construction looks like from far . the thicker triangles denote families of triangles that are very close to each other . the center part has only three points , @xmath83 , @xmath84 and @xmath85 . two of these , without loss of generality @xmath83 and @xmath85 , must receive the same color , say blue . after this we look more closely at the family @xmath86 that consists of the subfamilies @xmath87 , @xmath88 , @xmath89 and @xmath90 , see the bottom - left figure . unless the triangle @xmath91 is monochromatic , at least one of @xmath92 , @xmath93 , @xmath94 and @xmath95 must be blue . without loss of generality we suppose @xmath94 is blue . after this we look more closely at the family @xmath89 that consists of the triangles @xmath96 , @xmath97 , @xmath98 and @xmath99 , see the bottom - right figure . unless the triangle @xmath100 is monochromatic , at least one of @xmath101 , @xmath102 , @xmath103 and @xmath104 must be blue . but if @xmath105 is blue , then @xmath106 is monochromatic . this finishes the proof . in this section we investigate two - coloring geometric _ dynamic hypergraphs _ defined by intervals on a line . the vertices of a dynamic hypergraph are ordered and they `` appear '' in this order . knowing in advance the whole ordered hypergraph , our goal is to color the whole vertex set such that at all times any hyperedge restricted to the vertices that have `` arrived so far '' is non - monochromatic if it contains at least @xmath107 vertices that have arrived so far . this model is also called quasi - online in @xcite . the exact definitions are as follows .    for a hypergraph @xmath108 with an order on its vertices , @xmath109 , we define the _ dynamic closure _ of @xmath110 as the hypergraph on the same vertex set and with hyperedge set @xmath111 . a hypergraph with an order on its vertices is _ dynamic _ if it is its own dynamic closure . when @xmath112 is a hypergraph family , is the hypergraph family that contains all the dynamic closures of the family @xmath112 ( with all orderings of their vertex sets ) . a hypergraph is @xmath107-proper two - colorable if @xmath113 can be two - colored such that for every @xmath114 and @xmath115 if @xmath116 , then @xmath117 contains both colors . for a family of ( ordered ) hypergraphs , @xmath118 , we define the _ midriff _ of the family , @xmath119 , as the smallest number @xmath107 such that every ( ordered ) hypergraph in the family is @xmath107-proper two - colorable . [ obs : todynamic ] if for two non - ordered hypergraph families , @xmath120 is a subfamily of @xmath121 , then for their dynamic closures , @xmath122 is a subfamily of @xmath123 . now we look at the hypergraphs that we have studied in the earlier sections using the above terminology . : : :    vertices : a finite set of points from @xmath3 ; +    hyperedges : subsets of the vertices ( points ) contained in a quadrant    ( of the form @xmath30 ) . : : :    vertices : a finite set of quadrants ; +    hyperedges : subsets of the vertices ( quadrants ) that contain a point    from @xmath3 . : : :    vertices : a finite set of points from @xmath1 ; +    hyperedges : subsets of the vertices ( points ) contained in an octant    ( of the form    @xmath124 . : : :    vertices : a finite set of octants ; +    hyperedges : subsets of the vertices ( octants ) that contain a point    from @xmath1 .    we know that @xmath125 by definition and as we noted already in the previous section , in @xcite it was shown ( not using this terminology ) that decomposing octants is equivalent to its dual problem , i.e. , @xmath126and @xmath127 and that is the same as the ( ordered ) hypergraph family ( regarding the hypergraphs in it without the vertex orders ) . also , @xmath126 . summarizing :    equals and therefore + @xmath128 .    now we will define the hypergraph families that are the main topic of this section . the set of all intervals on the real line is denoted by @xmath129 . : : :    vertices : a finite point set ; +    hyperedges : subsets of the vertex points contained in an interval    @xmath130 . : : :    vertices : a finite set of intervals ; +    hyperedges : subsets of the vertices ( intervals ) containing a point    @xmath131 . : : :    vertices : a finite set of intervals ; +    hyperedges : subsets of the vertices ( intervals ) contained in an    arbitrary interval @xmath130 . : : :    vertices : a finite set of intervals ; +    hyperedges : subsets of the vertices ( intervals ) containing an    arbitrary interval @xmath130 . : : :    vertices : a finite set of intervals ; +    hyperedges : subsets of the vertices ( intervals ) intersecting an    arbitrary interval @xmath130 .    now we study the relations among the above five hypergraph families . by exchanging points with small enough intervals we get that the family is a subfamily of , and the family is a subfamily of , while both and are subfamilies of . observation  [ obs : todynamic ] implies several inequalities among their midriffs , for example , that @xmath132 . we are only aware of earlier papers studying the first two variants . it follows from a greedy algorithm that @xmath133 and @xmath134 . it was shown in @xcite that @xmath135 , and later this was generalized for @xmath14-colors in @xcite . it was also shown in @xcite that @xmath136 , and later this proof was simplified in @xcite . it is interesting to note that for the @xmath107-proper coloring problem there is a so - called _ semi - online _ algorithm , that can maintain an appropriate partial @xmath107-proper coloring of the points arrived so far , while it was shown in @xcite that no semi - online algorithm can exist for @xmath107-proper coloring . here we mainly study the other three hypergraph families . [ claim : ibiisi ] equals , thus equals .    by observation  [ obs : todynamic ] it is enough to prove the first statement . notice that in both and , we can suppose that the left endpoint of any vertex interval is to the left of the right endpoint of any vertex interval , as if we have a right endpoint such that the closest endpoint to its right is a left endpoint , then swapping them does not change the hypergraphs . thus , without loss of generality , there is a point that is in all the vertex intervals . instead of a line , imagine that the vertex intervals of a hypergraph of are the arcs of a circle such that none of them contains the bottommost point of the circle and all of them contains the topmost point . can be at most one larger than but we omit discussing this here . ] this is clearly equivalent to the version when the vertex intervals are on the line . similarly , we can imagine that the vertex intervals of a hypergraph of are the arcs of a circle such that none of them contains the topmost point of the circle and all of them contains the bottommost point . taking the complement of each arc transforms the families into each other , that is , in a hypergraph @xmath137 of an interval @xmath138 defines the hyperedge @xmath139 while in the hypergraph @xmath140 of whose vertices are the complements of the intervals in @xmath137 , the complement @xmath141 of @xmath138 defines the hyperedge @xmath142 . as @xmath143 if and only if @xmath144 , @xmath137 and @xmath140 are isomorphic , see figure  [ fig : circledual ] . [ lem : ibiisi ] and are both equal to , while is a family of subhypergraphs of hypergraphs from the above , and the same holds for the dynamic variants .    by observation  [ obs : todynamic ] , it is enough to prove the statements about the non - dynamic families . for an illustration for the proof , see figure  [ fig:3equi ] . recall that a quadrant is a set of the form @xmath30 for some apex @xmath34 . we can suppose that all points of the point set are in the north - eastern halfplane above the line @xmath145 defined by the function @xmath146 , i.e. , @xmath147 for every @xmath148 . for each point @xmath148 we define an interval , @xmath149 $ ] . quadrants that lie entirely below @xmath145 do not contain points from @xmath7 . for the quadrants with apex above @xmath145 , a quadrant whose apex is at @xmath40 contains the point @xmath39 if and only if @xmath150 contains @xmath151 . this shows that the hypergraphs in and in are the same . the equivalence of and already follows from proposition  [ claim : ibiisi ] , but we could give another proof in the above spirit , by supposing that for all points @xmath148 we have @xmath152 , moreover , that for every quadrant intersecting some of the points there is a quadrant containing the same set of points whose apex @xmath153 has @xmath154 . now for each point @xmath148 we can define the interval @xmath155 $ ] and proceed as before . note that this gives another proof for proposition  [ claim : ibiisi ] .    finally , taking a @xmath137 in , it is isomorphic to the subhypergraph of some @xmath140 in where in @xmath137 for all points @xmath148 we have @xmath152 and we take only the hyperedges corresponding to quadrants whose apex @xmath153 has @xmath156 . now for each point @xmath148 below @xmath145 we define @xmath155 $ ] , and for each point @xmath153 above @xmath145 we define @xmath157 $ ] , and proceed as before . this finishes the proof of the theorem . as it was shown in @xcite that @xmath158 , it follows that also @xmath159 @xmath160    quite surprisingly , we could not find a simple direct proof for the fact that is a family of subhypergraphs of hypergraphs from the families and .    from theorems  [ thm:9 ] and  [ thm : lower ] , and lemma  [ lem : ibiisi ] we obtain the following . @xmath161 and + @xmath162 our concluding diagram can be seen on figure  [ fig : diagram ] . most importantly , for octants we have @xmath163 and the same bound holds for the homothets and the translates of triangles . it seems to be in reach to determine these parameters exactly . for translates of convex @xmath164-gons for @xmath165 , the parameter @xmath107 might depend on the shape of the @xmath164-gon for @xmath164 fixed , and tends to infinity with @xmath164 @xcite . the upper and lower bounds @xcite are currently very far , already for a square . also , it is not known whether there exists an @xmath107 ( again depending on the convex @xmath164-gon ) such that any finite point set admits a two - coloring such that any homothet of the convex @xmath164-gon containing at least @xmath107 points is non - monochromatic , not even for the square . on the other hand , for any @xmath107 there is an @xmath107-fold covering by finitely many homothets of any convex @xmath164-gon ( for @xmath165 ) of some set that does not decompose to two coverings @xcite . a. asinowski , j. cardinal , n. cohen , s. collette , t. hackl , m. hoffmann , k. knauer , s. langerman , m. lason , p. micek , g. rote , t. ueckerdt , coloring hypergraphs induced by dynamic point sets and bottomless rectangles , in proceedings of the 13th algorithms and data structures symposium ( wads 2013 ) , lecture notes in computer science 8037 , 7384 , 2013 . j. cardinal , k. knauer , p. micek , t. ueckerdt , making triangles colorful , journal of computational geometry 4(1 ) : 240246 , 2013 . j. cardinal , k. knauer , p. micek , t. ueckerdt , making octants colorful and related covering decomposition problems , siam j. discrete math . 28(4 ) : 19481959 , 2014 . m. gibson and k. varadarajan , decomposing coverings and the planar sensor cover problem , discrete and computational geometry 46(2 ) : 313333 , 2011 . b. keszegh , d. plvlgyi , octants are cover decomposable into many coverings , computational geometry 47(5 ) : 585588 , 2014 . b. keszegh , d. plvlgyi , convex polygons are self - coverable , discrete and computational geometry 51(4 ) : 885895 , 2014 . i. kovcs , indecomposable coverings with homothetic polygons , discrete and computational geometry 53(4 ) : 817824 , 2015 . j. pach , d. plvlgyi , g. tth , survey on the decomposition of multiple coverings , in geometry , intuitive , discrete , and convex ( i. brny , k. j. brczky , g. fejes tth , j. pach eds . ) , bolyai society mathematical studies , vol . 24 , springer - verlag , 219257 , 2014 . d. plvlgyi , indecomposable coverings with concave polygons , discrete and computational geometry 44(3 ) : 577588 , 2010 . j. pach and d. plvlgyi , unsplittable coverings in the plane , to appear in the proceedings of the 41st international workshop on graph - theoretic concepts in computer science ( wg 2015 ) . d. plvlgyi and g. tth , convex polygons are cover - decomposable , discrete and computational geometry 43(3 ) : 483496 , 2010 .
in this note we improve our upper bound given in @xcite by showing that every @xmath0-fold covering of a point set in @xmath1 by finitely many translates of an octant decomposes into two coverings , and our lower bound by a construction for a @xmath2-fold covering that does not decompose into two coverings . 
 the same bounds also hold for coverings of points in @xmath3 by finitely many homothets or translates of a triangle . 
 we also prove that certain dynamic interval coloring problems are equivalent to the above question .