Spectral extremal hypergraphs without long Berge cycles

Abstract

Let $r\ge 3$ and $k\ge 2r+1$ be fixed integers. We determine, for all sufficiently large $n$, the maximum adjacency-tensor spectral radius of an $n$-vertex $r$-uniform hypergraph containing no Berge cycle of length at least $k$. Write $s=\left\lfloor\frac{k-1}{2}\right\rfloor$. If $k=2s+1$ is odd, the unique extremal hypergraph consists of all $r$-sets containing at most one vertex outside a fixed $s$-set. If $k=2s+2$ is even, one additionally includes all $r$-sets containing a fixed pair outside the $s$-set together with $r-2$ vertices inside it. Consequently, the maximum spectral radius is given as \[ \operatorname{spex}_r(n,k) = \left[ \binom{s}{r-1} \left(\frac{r-1}{s}\right)^{(r-1)/r} +o(1) \right]n^{(r-1)/r}. \]