Tur\'an Numbers of Linear Stars and Expanded Trees

Abstract

For an $r$-uniform hypergraph $\mathcal H$, let $\ex_r(n,\mathcal H)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no copy of $\mathcal H$. The $r$-uniform expansion $T^{(r)}$ of a tree $T$ is obtained by adding $r-2$ new vertices to each edge, with the added sets pairwise disjoint and disjoint from $V(T)$. In particular, the expansion of $K_{1,k}$ is the linear star $\mathcal L_k^r$, whose $k$ edges intersect pairwise in a single common vertex. Chung and Frankl (1987) determined $\ex_3(n,\mathcal L_k^3)$ and all extremal hypergraphs for every $k\ge3$ and sufficiently large $n$. We extend the odd-$k$ part of their result to every $r\ge5$, determining the exact Tur\'an number and the unique extremal hypergraph up to isomorphism for sufficiently large $n$. More generally, we extend this result to expansions of trees for which the minimum degree among the vertices in the smaller color class is odd and at least three. For every $r\ge5$ and sufficiently large $n$, we establish lower and upper bounds on their Tur\'an numbers. When no two vertices attaining this minimum degree in that color class have a common neighbor, we determine the exact Tur\'an number and the unique extremal hypergraph up to isomorphism. The proofs combine the delta-system method with stability arguments and a detailed analysis of extremal constructions.