Title

Edge distribution of graphs with few copies of a given graph

Abstract

We show that if a graph contains few copies of a given graph, then its edges are distributed rather unevenly. In particular, for all ε > 0 and r ≥ 2, there exist ζ= ζ(ε,r) > 0 and k = k(ε,r) such that, if n is sufficiently large and G = G(n) is a graph with fewer than ζnr r-cliques, then there exists a partition V(G) = ⊂i=0k Vi such that Vi = ⌊ n/k⌋ and e(Wi)< εVi2 for every i ∈ [k]. We deduce the following slightly stronger form of a conjecture of Erdos. For all c > 0 and r > 3, there exist ζ= ζ(c,r) > 0 and β= β(c,r) > 0 such that, if n is sufficiently large and G = G(n, [cn2]) is a graph with fewer than ζnr r-cliques, then there exists a partition V(G) = V1 ⊂ V2 with |V 1|= ⌊n/2⌋ and V2 = ⌈n/2⌉ such that e(V1,V2) > (1/2 + β)e(G).

Publication Title

Combinatorics Probability and Computing

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