Stability theorems for some Kruskal-Katona type results
Published in European Journal of Combinatorics Vol. 110, 2022
Recommended citation: Liu, Xizhi and Mukherjee, Sayan. "Stability theorems for some Kruskal-Katona type results." European J. Comb. 110 (2023) arXiv:2006.04848 (2020). https://doi.org/10.1016/j.ejc.2022.103666
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.