Extremal problems for suspensions of even cycles

Abstract

Given an integer $k\ge 2$ and a graph $F$, the $k$-uniform suspension ${\mathcal{S}}^{k}F$ is obtained by adjoining a fixed set of $k-2$ new vertices to every edge of $F$. In this paper, we study two extremal problems for suspensions of even cycles. Write $K_t^k$ for the $k$-uniform clique of order $t$. Let $\mathrm{ex}(n,{\mathcal{S}}^k{C}_{2\ell })$ and $\mathrm{ex}(n,K_t^k,{\mathcal{S}}^k{C}_{2\ell })$ denote the maximum numbers of edges and copies of $K_t^k$, respectively, in an ${\mathcal{S}}^k{C}_{2\ell }$-free $k$-uniform hypergraph on $n$ vertices. We prove that, for every $k\ge 2$ and infinitely many $n$, \[\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)=\frac{n^{k-1/2}}{(k+1)!}+O(n^{k-1}).\] This extends a folklore result for $k=2$ and, as an immediate consequence, yields the asymptotics of $\mathrm{ex}(n,{\mathcal{S}}^k{C}_4)$ for infinitely many $n$, previously established by Mubayi (for all $n$). Furthermore, for every $k\ge 2$ and $\ell \in \{3,5\}$, we determine the order of magnitude \[\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})=Θ(n^{k-1+1/\ell}).\] This generalizes both the classical graph case $k=2$ and a previous result of Mukherjee for $k=\ell =3$. The principal difficulty in both problems lies in constructing the lower bounds. Our construction for $\mathrm{ex}(n,K_{k+1}^k,{\mathcal{S}}^k{C}_4)$ incorporates a novel block-packing structure, which yields substantially more copies of $K_{k+1}^k$ than previously known constructions. For $\mathrm{ex}(n,{\mathcal{S}}^k{C}_{2\ell })$ with $\ell \in \{3,5\}$, we establish a natural $k$-uniform version of Wenger graphs, addressing the subtleties involved in lifting extremal graph constructions to suspensions. We also give applications of our results to Turán problems for simplicial complexes.