Degree versions of theorems on intersecting families via stability
Abstract
The matching number of a family of subsets of an -element set is the maximum number of pairwise disjoint sets. The families with matching number are called intersecting. The famous Erd\H os-Ko-Rado theorem determines the size of the largest intersecting family of -sets. Its generalization to the families with larger matching numbers, known under the name of the Erd\H{o}s Matching Conjecture, is still open for a wide range of parameters. In this paper, we address the degree versions of both theorems. More precisely, we give degree and -degree versions of the Erd\H{o}s-Ko-Rado and the Hilton-Milner theorems, extending the results of Huang and Zhao, and Frankl, Han, Huang and Zhao. We also extend the range in which the degree version of the Erd\H{o}s Matching conjecture holds.
Keywords
Cite
@article{arxiv.1810.00915,
title = {Degree versions of theorems on intersecting families via stability},
author = {Andrey Kupavskii},
journal= {arXiv preprint arXiv:1810.00915},
year = {2019}
}
Comments
This paper is one of two parts in which the paper arXiv:1710.02440 is split. The results in this part differ only slightly from Sections 5 and 6 of arXiv:1710.02440, but the presentation is improved