Turán's theorem for pseudo-random graphs

Abstract

The generalized Turán number 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 ex ( K m , H ) is ( 1 − 1 / ( χ ( H ) − 1 ) + o ( 1 ) ) ( m 2 ) , where o ( 1 ) → 0 as m → ∞ , 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 , β ) -bi-jumbled if | e G ( X , Y ) − q | X | | Y | | ⩽ β | X | | Y | for every two sets of vertices X , Y ⊂ V ( G ) . Here e G ( X , Y ) is the number of pairs ( x , y ) such that x ∈ X , y ∈ Y , and x y ∈ 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 δ > 0 there exists γ > 0 so that the following holds: any large enough m-vertex, ( q , γ q Δ + 1 / 2 m ) -bi-jumbled graph G satisfies ex ( G , H ) ⩽ ( 1 − 1 χ ( H ) − 1 + δ ) | E ( G ) | .