Improvements on induced subgraphs of given sizes

Given integers $m$ and $f$, let $S_n(m,f)$ consist of all integers $e$ such that every $n$-vertex graph with $e$ edges contains an $m$-vertex induced subgraph 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 of Erd\H{o}s, this was investigated by Erd\H{o}s, F\"uredi, Rothschild and S\'os twenty years ago. Their main result indicates that integers in $S_n(m,f)$ are rare 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 to provide 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 the constant $\frac12$ 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. We partially solve this conjecture (only leaving two open cases) by making progress on some constructions which are related to number theory. Our proofs are based on the research of Erd\H{o}s et. al and involve different arguments in number theory. We also discuss some related problems.