Spectral Tur\'an Type Problems on Cancellative Hypergraphs

TL;DR

This paper investigates the maximum spectral radius of cancellative 3-uniform hypergraphs, characterizes the extremal structures, and offers a spectral proof of a classical extremal combinatorics result.

Contribution

It determines the maximum p-spectral radius for cancellative hypergraphs and characterizes the extremal hypergraph, providing a spectral perspective on a known combinatorial extremal problem.

Findings

Maximum p-spectral radius achieved by balanced complete tripartite hypergraph.

Characterization of extremal hypergraph structure.

Alternative spectral proof of Bollobás's classical result.

Abstract

Let $G$ be a cancellative $3$-uniform hypergraph in which the symmetric difference of any two edges is not contained in a third one. Equivalently, a $3$-uniform hypergraph $G$ is cancellative if and only if $G$ is $\{F_4, F_5\}$-free, where $F_4 = \{abc, abd, bcd\}$ and $F_5 = \{abc, abd, cde\}$. A classical result in extremal combinatorics stated that the maximum size of a cancellative hypergraph is achieved by the balanced complete tripartite $3$-uniform hypergraph, which was firstly proved by Bollob\'as and later by Keevash and Mubayi. In this paper, we consider spectral extremal problems for cancellative hypergraphs. More precisely, we determine the maximum $p$-spectral radius of cancellative $3$-uniform hypergraphs, and characterize the extremal hypergraph. As a by-product, we give an alternative proof of Bollob\'as' result from spectral viewpoint.