Ramsey theory studies whether a *monochromatic* subgraph (more generally, structure) appears when we color the edges of a complete graph with some colors.
I wonder if the following type of question has been studied before, where monochromatic is replaced by bichromatic (which now I use in the sense of having *at most* two colors).



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We could define the *multicolor biRamsey number* of $G$, denoted by $biR\_k(G)$, as the least integer $n$ such that in any $k$-coloring of $K\_n$ there is a bichromatic copy of $G$.
Probably the simplest example is $biR\_3(K\_3)=5$, as a coloring of a complete graph has no bichromatic triangle if and only if every monochromatic component is a matching, and such a $3$-coloring exists for $K\_n$ if and only if $n\le 4.$


Obviously, we have $biR\_k(G)\le R\_k(G)$, but I don't see any general lower bound for $biR\_k$.
Has this parameter ever been studied?
(Possibly under a less brilliant name.)


Motivation: I would need something like in any $n$-coloring of a graph on $n$ vertices and $\Omega(n^2)$ edges there is a *trichromatic* $K\_{3,3}$, or something similar, where I have other conditions, like every color class is a matching.


Update: As you can see from the answers of Ilya and Jan, the answer to the motivation is negative, but I can produce other similar conditions, so I am still interested whether the problem has been studied.



How many colors do we need to avoid bichromatic triangles?
Problems of this variety have been studied, beginning with a paper of Erdős and Gyárfás, 'A variant of the classical Ramsey problem'. In that paper, they define a function $f(n,p,q)$ to be the smallest number of colours $k$ needed to produce a $k$-colouring of the edges of $K\_n$ such that every $K\_p$ contains at least $q$ distinct colours. The study of this function and its variants (such as replacing $K\_p$ with a general graph $H$) leads to a great number of interesting questions. For example, even the behaviour of $f(n,4,3)$ is poorly understood.


