Full subgraphs

Let G = (V,E) be a graph of density p on n vertices. Following Erdős, Luczak and Spencer [11], an m-vertex subgraph H of G is called full if H has minimum degree at least p(m − 1). Let f(G) denote the order of a largest full subgraph of G. If p ( n 2 ) is a positive integer, define fp(n) = min{f(G) : |V (G)| = n, |E(G)| = p ( n 2 ) }. Erdős, Luczak and Spencer [11] proved that if p = 1 2 , then for n ≥ 2, (2n) 1 2 − 2 ≤ fp(n) ≤ 4n 23 (logn) 13 . In this paper, we show that for infinitely many p near the elements of { 1 2 , 2 3 , 3 4 , . . . }, fp(n) = Θ(n 2 3 ), and more generally that for all p : n− 2 3 < p < 1− n− 15 we have fp(n) = Ω((1 − p) 13n 2 3 ). As an ingredient of the proof, we show that every graph G on n vertices has a subgraph H on m vertices, ⌊ r ⌋ ≤ m ≤ ⌈ r ⌉+1 vertices such that for every v ∈ V (H), the degree of v in H is at least 1/r times its degree in G. Finally, we discuss full subgraphs of random and pseudorandom graphs, and several open problems.