Large nearly regular induced subgraphs

For a real c ≥ 1 and an integer n, let f(n, c) denote the maximum integer f such that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in particular, every graph on n vertices contains a regular induced subgraph on at least f(n, 1) vertices. The problem of estimating f(n, 1) was posed long ago by Erdös, Fajtlowicz, and Staton. In this paper we obtain the following upper and lower bounds for the asymptotic behavior of f(n, c): (i) For fixed c > 2.1, n^1-0(1/c) ≤ f(n, c) ≤ O(cn/ logra). (ii) For fixed c = 1 + η with η > 0 sufficiently small, f(n,c) ≥ nΩ(η² /ln(1/η)). (iii) Ω(lnra) ≤ f(n, 1) ≤ O(n^1/2 ln^3/4ra). An analogous problem for not necessarily induced subgraphs is briefly considered as well.