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Given a parameter $k$, and a graph $G$ with $O(k^2)$ vertices that has a maximum clique with $\le k$ vertices, I want to investigate the maximum number of colors $C(k)$ needed to properly color $G$, i.e $\chi(G)\le C(k)$.

Using Mycielski's graphs, one can show that this number should be at least $k+\log k$.

It is known that for any integer $c$ there is a triangle-free graph that has chromatic number $c$. However in all these constructions, as far as I know, the size of these graphs is a rapidly increasing function of $c$.

I conjecture that $C(k)=\Theta(k)$ and I would be really grateful for any helpful references related to this question.

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    $\begingroup$ Do you mean the size (number of edges) or the order (number of vertices) to be $O(k^2)$? $\endgroup$
    – Gamow
    Mar 3, 2020 at 17:30
  • $\begingroup$ Sorry for the confusion, I actually meant "order". I make an edit $\endgroup$ Mar 3, 2020 at 23:32

2 Answers 2

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Uniformly random $k^2$-vertex graphs have clique size $O(\log k)$, well under $k$, and independent set size also $O(\log k)$, implying that their chromatic number is $\Omega(k^2/\log k)$.

As for $k^2$-vertex triangle-free graphs, their chromatic number can be $\Theta(k/\sqrt{\log k})$ (and not higher); see Kim, Jeong Han (1995), "The Ramsey number $R(3,t)$ has order of magnitude $t^2/\log t$", Random Structures & Algorithms 7 (3): 173–207, doi:10.1002/rsa.3240070302

If by "size" you mean edges rather than vertices, then the maximum chromatic number of triangle-free $k^2$-edge graphs is $\Theta((k/\log k)^{2/3})$. See Nilli, A. (2000), "Triangle-free graphs with large chromatic numbers", Discrete Mathematics 211(1–3): 261–262, doi:10.1016/S0012-365X(99)00109-0.

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  • $\begingroup$ To add to this, for David's third comment, it was actually Poljak and Tuza (1994) who first showed this (modulo Kim's result): S. Poljak and Z. Tuza. Bipartite subgraphs of triangle-free graphs. SIAM J. Discrete Math., 7(2):307–313, 1994. Moreover, Gimbel and Thomassen (2000) were contemporaneous to Noga: J. Gimbel and C. Thomassen. Coloring triangle-free graphs with fixed size. Discrete Math., 219(1-3):275–277, 2000. $\endgroup$
    – RJK
    Apr 20, 2020 at 17:26
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In addition to the Mycielski graphs, I guess you only need an upper bound on the chromatic number, as a function of the graph size. Such a bound is mentioned e.g. here, showing that a graph of size $O(k^2)$ has chromatic number $O(k)$, which proves that $C(k) \in \Theta(k)$.

EDIT: This answer is invalidated by an update to the original question.

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