Bounded chromatic number of graphs with small clique number and large minimum degree

Source abstract

We prove that every triangle-free graph with minimum degree at least $\frac{n}{3}$ is $4$-colorable and thereby settle a problem of Brandt and Thomassé (2005) at the threshold $\frac{n}{3}$. The number four is best possible. For a positive integer-valued function $f(n)=o(n)$, we relate the chromatic number of $f(n)$-vertex subgraphs of the Kneser graph $KG(n,f(n))$ to that of triangle-free graphs with minimum degree at least $\frac{n}{3}-f(n)$. Consequently, for every $0 0$, and for all sufficiently large $n$, every $n$-vertex triangle-free graph with minimum degree at least $\frac{n}{3}-n^{1-δ}$ has chromatic number at most $10^{391}+1+\left\lceil{(1+\varepsilon)(1-δ)}/δ\right\rceil$. We also show that every sufficiently large $n$-vertex maximal triangle-free graph with minimum degree at least $\frac{n}{3}-f(n)$ and chromatic number at least $10^{391}$ contains a bipartite subgraph with parts of orders $\frac{n}{3}-O(f(n))$ and $\frac{2n}{3}-O(f(n))$; the remaining induced subgraph admits a homomorphism to $KG(\frac{n}{3}-O(f(n)),O(f(n)))$. Finally, we connect maximal $K_r$-free graphs with minimum degree at least $\frac{2r-5}{2r-3}n-f(n)$ to $K_{r-1}$-free graphs and extend these results to $K_r$-free graphs. Our proofs employ the recent strong Brandt--Thomassé theorem of Łuczak, Polcyn, and Reiher.