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Given integers $m$ and $f$, let $S_n(m,f)$ consist of all integers $e$ suchthat every $n$-vertex graph with $e$ edges contains an $m$-vertex inducedsubgraph with $f$ edges, and let $\sigma(m,f)=\limsup_{n\rightarrow\infty}|S_n(m,f)|/\binom{n}{2}$. As a natural extension of an extremal problem ofErd\H{o}s, this was investigated by Erd\H{o}s, F\"uredi, Rothschild and S\'ostwenty years ago. Their main result indicates that integers in $S_n(m,f)$ arerare for most pairs $(m,f)$, though they also found infinitely many pairs$(m,f)$ whose $\sigma(m,f)$ is a fixed positive constant. Here we aim toprovide some improvements on this study. Our first result shows that$\sigma(m,f)\leq \frac12$ holds for all but finitely many pairs $(m,f)$ and theconstant $\frac12$ cannot be improved. This answers a question of Erd\H{o}s et.al. Our second result considers infinitely many pairs $(m,f)$ of special forms,whose exact values of $\sigma(m,f)$ were conjectured by Erd\H{o}s et. al. Wepartially solve this conjecture (only leaving two open cases) by makingprogress on some constructions which are related to number theory. Our proofsare based on the research of Erd\H{o}s et. al and involve different argumentsin number theory. We also discuss some related problems.

Improvements on induced subgraphs of given sizes