Stability theorems for some Kruskal-Katona type results

TL;DR

This paper establishes stability theorems for certain classes of hypergraphs, showing that hypergraphs close to extremal size are structurally similar to known extremal configurations, extending classical combinatorial results.

Contribution

It provides new stability results for cancellative hypergraphs and hypergraphs without clique expansions, generalizing Kruskal-Katona type theorems.

Findings

Stability results for cancellative hypergraphs

Stability results for hypergraphs without clique expansions

Structural proximity to extremal configurations

Abstract

The classical Kruskal-Katona theorem gives a tight upper bound for the size of an $r$-uniform hypergraph $\mathcal{H}$ as a function of the size of its shadow. Its stability version was obtained by Keevash who proved that if the size of $\mathcal{H}$ is close to the maximum, then $\mathcal{H}$ is structurally close to a complete $r$-uniform hypergraph. We prove similar stability results for two classes of hypergraphs whose extremal properties have been investigated by many researchers: the cancellative hypergraphs and hypergraphs without expansion of cliques.