Graphs with induced subgraphs of any number of edges

Problem (Proposed by Erdös and McKay\( ^{[1]}\))($100)

Let \(f(n,c)\) denote the largest integer $m$ such that any graph \(G\) on \(n\) vertices containing no clique or independent set of size \(c \log n\) must contain an induced subgraph with exactly \(i\) edges for each \(i\), \(0 < i \leq m\). Prove or disprove that \(f(n,c) \geq \epsilon n^2\).

McKay wrote \( ^{[1]}\), ``It is easy to get bounds of the form \( f(n,c) \geq c' \log n\), and Paul had a more complicated way to prove the bound \( f(n) \geq c' (\log n)^2\), but I cannot remember it."

Calkin, Frieze and McKay \( ^{[2]}\) proved that a random graph with \( pn^2\) edges, for constant \( p\), contains an induced subgraph with exactly \( i\) edges for each \( i\), for \( i\) ranging from 0 up to \( (1-\epsilon) pn^2\).

Bibliography
1 P. Erdös, Some of my favourite problems in number theory, combinatorics, and geometry, Combinatorics Week (Portuguese) (São Paulo, 1994), Resenhas 2 (1995), 165-186.

2 N. Calkin, A. Frieze and B. D. McKay, On subgraph sizes of random graphs, Combinatorics, Probability and Computing 1 (1992), 123-134.