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In this paper, we investigate the minimum number of triangles, denoted by $t(n,k)$, in $n$-vertex $k$-regular graphs, where $n$ is an odd integer and $k$ is an even integer. The well-known Andr\'asfai-Erd\H{o}s-S\'os Theorem has established that $t(n,k)>0$ if $k>\frac{2n}{5}$. In a striking work, Lo has provided the exact value of $t(n,k)$ for sufficiently large $n$, given that $\frac{2n}{5}+\frac{12\sqrt{n}}{5}

Let $G$ be a $K_p$-free graph. We say $e$ is a $K_p$-saturating edge of $G$ if $e\notin E(G)$ and $G+e$ contains a copy of $K_p$. Denote by $f_p(n, e)$ the minimum number of $K_p$-saturating edges that an $n$-vertex $K_p$-free graph with $e$ edges can have. Erd\H{o}s and Tuza conjectured that $f_4(n,\lfloor n^2/4\rfloor+1)=\left(1 + o(1)\right)\frac{n^2}{16}.$ Balogh and Liu disproved this by showing $f_4(n,\lfloor n^2/4\rfloor+1)=(1+o(1))\frac{2n^2}{33}$. They believed that a natural generalization of their construction for $K_p$-free graph should also be optimal and made a conjecture that $f_{p+1}(n,ex(n,K_p)+1)=\left(\frac{2(p-2)^2}{p(4p^2-11p+8)}+o(1)\right)n^2$ for all integers $p\ge 3$. The main result of this paper is to confirm the above conjecture of Balogh and Liu.

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.

Extremal problems on the $4$-cycle $C_4$ played a heuristic important role in the development of extremal graph theory. A fundamental theorem of F\"uredi states that the Tur\'an number $ex(q^2+q+1, C_4)\leq \frac12 q(q+1)^2$ holds for every $q\geq 14$, which matches with the classic construction of Erd\H{o}s-R{\'e}nyi-S\'os and Brown from finite geometry for prime powers $q$. Very recently, we obtained the first stability result on F\"uredi's theorem, by showing that for large even $q$, every $(q^2+q+1)$-vertex $C_4$-free graph with more than $\frac12 q(q+1)^2-0.2q$ edges must be a spanning subgraph of a unique polarity graph. Using new technical ideas in graph theory and finite geometry, we strengthen this by showing that the same conclusion remains true if the number of edges is lowered to $\frac12 q(q+1)^2-\frac12 q+o(q)$. Among other applications, this gives an immediate improvement on the upper bound of $ex(n,C_4)$ for infinitely many integers $n$. A longstanding conjecture of Erd\H{o}s and Simonovits states that every $n$-vertex graph with $ex(n,C_4)+1$ edges contains at least $(1+o(1))\sqrt{n}$ 4-cycles. We proved an exact result and confirmed Erd\H{o}s-Simonovits conjecture for infinitely many integers $n$. As the second main result of this paper, we further characterize all extremal graphs for which achieve the $\ell$th least number of copies of $C_4$ for any fixed positive integer $\ell$. This can be extended to more general settings and provides enhancements on the understanding of the supersaturation problem of $C_4$.