Full subgraphs

Let $G=(V,E)$ be a graph of density $p$ on $n$ vertices. Following Erd\H{o}s, \L uczak and Spencer, an $m$-vertex subgraph $H$ of $G$ is called {\em 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 $p\binom{n}{2}$ is a non-negative integer, define [ f(n,p) = \min{f(G) : \vert V(G)\vert = n, \ \vert E(G)\vert = p\binom{n}{2} }.] Erd\H{o}s, \L uczak and Spencer proved that for $n \geq 2$, [ (2n)^{{\frac{1}{2}}} - 2 \leq f(n, {\frac{1}{2}}) \leq 4n^{{\frac{2}{3}}(\log} n)^{{\frac{1}{3}}.]} In this paper, we prove the following lower bound: for $n^{{-\frac{2}{3}}} <p_n <1-n^{{-\frac{1}{7}}$,} [ f(n,p) \geq \frac{1}{4}(1-p)^{{\frac{2}{3}}n{\frac{2}{3}}} -1.] Furthermore we show that this is tight up to a multiplicative constant factor for infinitely many $p$ near the elements of ${\frac{1}{2},\frac{2}{3},\frac{3}{4},\dots}$. In contrast, we show that for any $n$-vertex graph $G$, either $G$ or $G^c$ contains a full subgraph on $\Omega(\frac{n}{\log n})$ vertices. Finally, we discuss full subgraphs of random and pseudo-random graphs, and several open problems.