Extremal problems for suspensions of even cycles

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 $\mathrm{e}\mathrm{x}(n,{\mathcal{S}}^k{C}_{2\mathrm{\ell }})$ and $\mathrm{e}\mathrm{x}(n,K_t^k,{\mathcal{S}}^k{C}_{2\mathrm{\ell }})$ denote the maximum numbers of edges and copies of $K_t^k$, respectively, in an ${\mathcal{S}}^k{C}_{2\mathrm{\ell }}$-free $k$-uniform hypergraph on $n$ vertices. We prove that, for every $k\ge 2$ and infinitely many $n$, $$\mathrm{e}\mathrm{x}(n,K_{k+1}^k,{\mathcal{S}}^k{C}_4)=\frac{n^{k-1\mathrm{/}2}}{(k+1)!}+O(n^{k-1})\mathrm{.}$$ This extends a folklore result for $k=2$ and, as an immediate consequence, yields the asymptotics of $\mathrm{e}\mathrm{x}(n,{\mathcal{S}}^k{C}_4)$ for infinitely many $n$, previously established by Mubayi (for all $n$). Furthermore, for every $k\ge 2$ and $\mathrm{\ell }\in \{3,5\}$, we determine the order of magnitude $$\mathrm{e}\mathrm{x}(n,{\mathcal{S}}^k{C}_{2\mathrm{\ell }})=\mathrm{\Theta }(n^{k-1+1\mathrm{/}\mathrm{\ell }})\mathrm{.}$$ This generalizes both the classical graph case $k=2$ and a previous result of Mukherjee for $k=\mathrm{\ell }=3$. The principal difficulty in both problems lies in constructing the lower bounds. Our construction for $\mathrm{e}\mathrm{x}(n,K_{k+1}^k,{\mathcal{S}}^k{C}_4)$ incorporates a novel block-packing structure, which yields substantially more copies of $K_{k+1}^k$ than previously known constructions. For $\mathrm{e}\mathrm{x}(n,{\mathcal{S}}^k{C}_{2\mathrm{\ell }})$ with $\mathrm{\ell }\in \{3,5\}$, 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.