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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 $σ(m,f)=\limsup_{n\rightarrow\infty} |S_n(m,f)|/\binom{n}{2}$. As a natural extension of an extremal problem of Erdős, this was investigated by Erdős, Füredi, Rothschild and Sós 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 $σ(m,f)$ is a fixed positive constant. Here we aim to provide some improvements on this study. Our first result shows that $σ(m,f)\leq \frac12$ holds for all but finitely many pairs $(m,f)$ and the constant $\frac12$ cannot be improved. This answers a question of Erdős et. al. Our second result considers infinitely many pairs $(m,f)$ of special forms, whose exact values of $σ(m,f)$ were conjectured by Erdős 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 Erdős et. al and involve different arguments in number theory. We also discuss some related problems.