Extremal problems for suspensions of even cycles

Given an integer $k\geq2$ and a graph $F$, the $k$-uniform suspension $\mathcal{S}^kF$ 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 $K^k_t$ for the $k$-uniform clique of order $t$. Let $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ and $\mathrm{ex}(n,K^k_t,\mathcal{S}^kC_{2\ell})$ denote the maximum numbers of edges and copies of $K^k_t$, respectively, in an $\mathcal{S}^kC_{2\ell}$-free $k$-uniform hypergraph on $n$ vertices. We prove that, for every $k\geq2$ 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 $\mathrm{ex}(n,\mathcal{S}^kC_4)$ for infinitely many $n$, previously established by Mubayi (for all $n$). Furthermore, for every $k\geq2$ 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 $\mathrm{ex}(n,K^{k}_{k+1},\mathcal{S}^kC_4)$ incorporates a novel block-packing structure, which yields substantially more copies of $K^k_{k+1}$ than previously known constructions. For $\mathrm{ex}(n,\mathcal{S}^kC_{2\ell})$ 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.