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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 Km on m vertices, the value of ex(Km, H) is (1−1/(χ(H)−1)+o(1)) ` ´ m, where o(1) → 0 as m → ∞, 2 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 [33]. A graph G is (q, β)-bi-jumbled if ˛ p ˛eG(X, Y) − q|X||Y | ˛ ≤ β |X||Y | for every two sets of vertices X, Y ⊂ V (G). Here eG(X, Y) is the number of pairs (x, y) such that x ∈ X, y ∈ Y, and xy ∈ 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 ∆ and for any δ> 0 there exists γ> 0 so that the following holds: any large enough m-vertex, (q, γq ∆+1/2 m)-bi-jumbled graph G satisfies ex(G, H)

Year: 2006

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