Tur\'{a}n numbers of general hypergraph star forests

TL;DR

This paper extends the understanding of Turán numbers for hypergraph star forests, generalizing previous results to broader classes of hypergraph configurations and providing new bounds for these combinatorial structures.

Contribution

It generalizes existing Turán number results from star forests with identical stars to arbitrary hypergraph star forests, broadening the scope of extremal hypergraph theory.

Findings

Established new upper bounds for Turán numbers of general hypergraph star forests.

Extended previous results to non-uniform hypergraph star forests.

Connected hypergraph star forest Turán numbers to classical star forest results.

Abstract

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs, and let $H$ be an $r$-uniform hypergraph. Then $H$ is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$ as a subhypergraph. The Tur\'{a}n number of $\mathcal{F}$, denoted by $ex_r(n,\mathcal{F})$, is the maximum number of hyperedges in an $\mathcal{F}$-free $n$-vertex $r$-uniform hypergraph. Our current results are motivated by earlier results on Tur\'{a}n numbers of star forests and hypergraph star forests. In particular, Lidick\'{y}, Liu and Palmer [Electron. J. Combin. 20 (2013)] determined the Tur\'{a}n number $ex(n,F)$ of a star forest $F$ for sufficiently large $n$. Recently, Khormali and Palmer [European. J. Combin. 102 (2022) 103506] generalized the above result to three different well-studied hypergraph settings, but restricted to the case that all stars in the hypergraph star forests are identical. We further generalize these results to general hypergraph star forests.