In 2016, Reiher's clique density theorem determined the minimum number of copies of $K_t$ in a graph with a prescribed edge density. In this paper, we investigate its local version and prove a local clique density theorem in $H$-free graphs as follows. For integers $r$ and $t$ with $2\leq t\leq r-1$, any $r$-chromatic graph $H$, any real numbers $\gamma$ and $\alpha$ with $\frac{t-2}{2(t-1)}\leq\gamma\leq \frac{r-2}{2(r-1)}$ and $0\leq\alpha\leq 1$, we determine the maximum value $\beta:=\beta(r,t,\alpha,\gamma)$ such that for every $n$-vertex $H$-free graph $G$ with at least $\gamma n^2$ edges, every $\lceil\alpha n\rceil$-vertex subset in $G$ contains at least $(\beta-o(1))n^{t}$ copies of $K_t$. In particular, when $H=K_r$, every $\lceil\alpha n\rceil$-vertex subset contains at least $\lfloor\beta n^t\rfloor$ copies of $K_t$, which is an exact bound. For suitable choices of $\alpha$ and $\gamma$, namely, those for which all part ratios in the corresponding extremal construction are rational, this bound is attained for infinitely many values of $n$.
For graphs $H$ and $F$, let $ex(n,H,F)$ be the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. We study this problem when $H$ is a clique and $F=T_t$which is a fixed tree on $t$ vertices. The Erd\H{o}s--S\'{o}s conjecture concerns the value of $ex(n,K_2, T_t)$. Gerbner and Palmer proposed a more genera...
Given graphs $H$ and $F$, the generalized Tur\'{a}n number ${\rm ex}(n,H,F)$ is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. Alon and Shikhelman (J. Combin. Theory Ser. B, 2016) initiated the systematic study of generalized Tur\'{a}n problems. Let $T$ be a tree on $k$ vertices, and write $n=a(k-...
The codegree Tur\'an density $\pi_{\mathrm{co}}(F)$ is the supremum over all $\gamma \in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $\gamma n$ edges. Ding, Lamaison, Liu, Wang, and Yang (JLMS, 2025) studied the p...
Let $F$ be a connected color-critical graph with $\chi(F)=r+1\ge4$, let $S_{n,\Delta}^{(r)}=(n-\Delta)K_1\vee T(\Delta,r-1)$. We determine the graph of maximum adjacency spectral radius among all $n$-vertex $F$-free graphs with prescribed maximum degree $\Delta$. There is a constant $s_F\in[0,1)$ such that, for all suf...
A graph $\Gamma$ is $(c,t)$-sparse for $c>0$ and $t \ge 1$ if for every pair of vertex subsets $A, B \subseteq V(\Gamma)$ with $|A|, |B| \ge t$, the number of edges $e(A,B)$ between them satisfies $ e(A,B) \le (1 - c)|A||B|$. In this paper, we prove that for every integer $\ell\ge2$, there are $\varepsilon>0, C, C'>0$...
Adam Džavoronok, Ole Gabsdil, Alexander Mylet et al.· 1 citation
Let $X_H$ denote the number of copies of a fixed graph $H$ in $G_{n, p}$. Gilmer and Kopparty conjectured that $X_H$ satisfies a local central limit theorem (LCLT) provided that $H$ is connected, $p \gg n^{-1/m(H)}$, and $n^2 (1-p) \gg 1$, where $m(H)$ is the maximum density. Following the work of Berkowitz, Sah and Sa...
Asaf Cohen Antonir, Ilay Hoshen, M. Zhukovskii· 1 citation
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
