For fixed graphs $H$ and $F$, let $\ex(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. In this note, we prove the generalized rational exponents conjecture, posed by Gerbner and Palmer, showing that for every rational number $\alpha\ge1$, there exist fixed graphs $H_\alpha$ and $F_\alpha$ such that \[ \ex(n,H_\alpha,F_\alpha)=\Theta(n^\alpha). \] Furthermore, the counting graph $H_\alpha$ can always be chosen connected with diameter at most $3$. Our argument hinges on a localization--compression--shift framework, which transforms the Bukh--Conlon finite family construction for edges into a generalized Tur\'an problem setting with a single forbidden graph.
Given a graph $H$, the extremal number $ex(n,H)$ is the maximum number of edges in an $n$-vertex graph not containing $H$ as a subgraph. The well-known rational exponents conjecture of Erd\H{o}s and Simonovits states that for any rational $\gamma\in (1,2)$ there exists a single bipartite graph $H$ satisfying $ex(n,H)=\Theta(n^\gamma)$. Among other results, the conjecture has been verified for all $\gamma=1+a/b$, where $b>a^2$, by Jiang and Qiu and for all $\gamma=2-a/b$, where $b>\max\{a, (a-1)^2\}$, by Conlon and Janzer. In this paper, we establish the rational exponents conjecture for many $\gamma$ near the center of the interval, namely, for all $\gamma=1+\frac{rt-1}{2rt+2r}$, where $r,t$ are natural numbers satisfying $t\geq 2$, $r\geq 2t+3$.
Tao Jiang, Sean Longbrake, Liana Yepremyan· 0 citations
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. Recently, Gao, Wu and Xue (J. Graph Theory, 2026) asked whether every graph $F$ with chromatic number $\chi(F)=r\geq3$ and treewidth ${\rm tw}(F)\geq r$ satisfies ${\rm ex}(n,K_r,F)=\Omega(n^{r-1})$. In this note, we give a negative answer to this question for every $r\geq3$. More precisely, we prove that the graph $F_r=K_{r-3}\vee H$, where $H$ is obtained from $K_4$ by subdividing one edge once, satisfies $\chi(F_r)={\rm tw}(F_r)=r$ and \[ n^{r-1}e^{-O(\sqrt{\log n})}\leq {\rm ex}(n,K_r,F_r)=o(n^{r-1}). \] This result also disproves Conjecture 6.3 in the recent survey of Gerbner and Palmer (Electron. J. Combin., 2026).
For graphs $G$ and $H$, let $\mathbf N(G,H)$ denote the number of unlabeled, not necessarily induced copies of $H$ in $G$, and let $\mathbf N_{\mathcal P}(n,H)$ be the maximum of $\mathbf N(G,H)$ over all $n$-vertex planar graphs $G$. We prove that, for every fixed integer $m\geq 3$, $$\mathbf N_{\mathcal P}(n,C_{2m+1})=2m\left(\frac{n}{m}\right)^m+O_m\!\left(n^{m-1/5}\right).$$ The proof uses a sharp weighted cycle--path inequality for edge probability measures on finite complete graphs. This strengthens a conjecture of Heath, Martin, and Wells and, together with their reduction lemma, yields the stated asymptotic formula.
For a graph $H$ with $3\mid e(H)$, the zero-sum Ramsey number $R(H,\Z_3)$ is the least integer $N$ such that every labeling of the edges of $K_N$ by elements of $\Z_3$ contains a copy of $H$ whose edge labels sum to zero. We determine the last previously unresolved infinite family in the complete-graph case modulo $3$. More precisely, we prove that \(R(K_n,\Z_3)=n+3\) for every $n\ge 10$ satisfying $n\equiv 1\pmod 3$. Consequently, for $k\ge 1$, \(R(K_{9k+7},\Z_3)=9k+10\), resolving a problem of Caro and Mifsud.
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 general conjecture: if $n=\alpha(t-1)+\beta$ and $0\le\beta\le t-2$, then the graph $\alpha K_{t-1}\sqcup K_\beta$ maximizes the number of $r$-cliques among all $n$-vertex $T_t$-free graphs for every $3\le r\le t-2$. We show that this conjecture holds for $T_t$ having at least $t-r$ leaves with a common parent, which contains the star case as a special case and recovers the sharp clique-counting result conjectured by Gan, Loh and Sudakov and proved by Chase and Chao and Dong. We also study the clique-spectral analogue. Under the same leaf-bunch condition, every $T_t$-free graph $G$ satisfies $\rho_r(G)\le\binom{t-2}{r-1}$, with equality, for $n\ge t-1$, if and only if $K_{t-1}$ is a component of $G$. Furthermore, we prove the conjecture for $r=t-d$ whenever $d\ge2$ and $t\ge d^2-d+3$, while the case $d=1$ is determined exactly for every $t$. For $d\ge2$ and $t\ge d^2-d+3$, every $T_t$-free graph $G$ satisfies $\rho_{t-d}(G)\le\rho_{t-d}(K_{t-1})$, with equality characterized by the presence of a $K_{t-1}$-component. Our method is designed for relatively large cliques. In the leaf-poor case, after deleting edges that lie in no $(t-d)$-clique, we study the intersection relation among $(t-d)$-cliques and show that its equivalence classes induce the nontrivial clique-supported components; furthermore, we show each non-trivial component has at most $t-1$\) vertices. In the complementary leaf-rich case, a leaf-bunch criterion reduces the clique-counting problem to the sharp bounded-maximum-degree clique theorem.
For a finite group $H$, let $\nu(H)$ denote the maximum order of a nilpotent subgroup of $H$. We prove that every finite solvable transitive permutation group $P$ on a finite set $\Omega$ has a subset $\Delta\subseteq\Omega$ such that $|P:P_\Delta|\ge\nu(P_\Delta)$. We also prove that if a finite solvable group $H$ acts faithfully and completely reducibly on a finite module $V$, then some $x\in V$ satisfies $|H:H_x|\ge\nu(H_x)$. Consequently, we settle Gluck's well-known conjecture, open since 1985: every finite solvable group $G$ satisfies $|G:\mathbf{F}(G)|\le b(G)^2$, where $\mathbf{F}(G)$ is the largest normal nilpotent subgroup of $G$ and $b(G)$ is the largest degree of an irreducible complex character of $G$.
Baoyu Zhang· 0 citations
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