Spectral Turán problems for complete bipartite expansions in linear hypergraphs

Source abstract

An $r$-uniform hypergraph is linear if any two edges share at most one vertex. For a family $\mathcal F$ of $r$-graphs, let $\text{ex}^{\text{lin}}_r(n,\mathcal F)$ and $\text{spex}^{\text{lin}}_r(n,\mathcal F)$ denote, respectively, the maximum number of edges and the maximum spectral radius of an $n$-vertex $\mathcal F$-free linear $r$-graph. In this paper, we show that for $t\ge s\ge3$, $\text{spex}^{\text{lin}}_r(n,K_{s,t}^{+})\le \frac{(t-s+1)^{1/s}}{r-1}n^{1-1/s}+o(n^{1-1/s})$, where $K_{s,t}^{+}$ is the $r$-graph obtained from the complete bipartite graph $K_{s,t}$ by enlarging each edge with $r-2$ new vertices. This answers a problem of She, Fan and Kang in a stronger form. Moreover, our result implies that $\text{ex}^\text{lin}_r(n,K_{s,t}^{+}) \le \frac{(t-s+1)^{1/s}}{r(r-1)}n^{2-1/s}+o(n^{2-1/s})$, which improves the leading coefficient in the corresponding bound of Gao and Chang. We also give a construction which implies that $\text{spex}^{\text{lin}}_r(n,K_{s,t}^{+})=Θ(n^{1-1/s})$ and $\text{ex}^\text{lin}_r(n,K_{s,t}^{+})=Θ(n^{2-1/s})$ for $s\ge 4r-6$ and $t>(s-1)!$.