Turán's theorem for pseudo-random graphs

Abstract

The generalized Turán number $\mathrm{ex}(G,H)$ of two graphs G and H is the maximum number of edges in a subgraph of G not containing H. When G is the complete graph $K_m$ on m vertices, the value of $\mathrm{ex}(K_m,H)$ is $(1-1/(\chi (H)-1)+o(1))\left(\begin{array}{c}\hfill m\hfill \\ \hfill 2\hfill \end{array}\right)$, where $o(1)\to 0$ as $m\to \infty$, by the Erdős–Stone–Simonovits theorem.

In this paper we give an analogous result for triangle-free graphs H and pseudo-random graphs G. Our concept of pseudo-randomness is inspired by the jumbled graphs introduced by Thomason [A. Thomason, Pseudorandom graphs, in: Random Graphs '85, Poznań, 1985, North-Holland, Amsterdam, 1987, pp. 307–331. MR 89d:05158]. A graph G is $(q,\beta )$-bi-jumbled if

$$|e_G(X,Y)-q|X||Y||⩽\beta \sqrt{|X||Y|}$$

for every two sets of vertices $X,Y\subset V(G)$. Here $e_G(X,Y)$ is the number of pairs $(x,y)$ such that $x\in X$, $y\in Y$, and $xy\in E(G)$. This condition guarantees that G and the binomial random graph with edge probability q share a number of properties.

Our results imply that, for example, for any triangle-free graph H with maximum degree Δ and for any $\delta >0$ there exists $\gamma >0$ so that the following holds: any large enough m-vertex, $(q,\gamma q^{\Delta +1/2}m)$-bi-jumbled graph G satisfies

$$\mathrm{ex}(G,H)⩽(1-\frac{1}{\chi (H)-1}+\delta )|E(G)|.$$