Degree versions of theorems on intersecting families via stability

Abstract

For a family of subsets of an n-element set, its matching number is the maximum number of pairwise disjoint sets. Families with matching number 1 are called intersecting. The famous Erdős–Ko–Rado theorem determines the size of the largest intersecting family of k-sets. The problem of determining the largest family of k-sets with matching number s>1 is known under the name of the Erdős Matching Conjecture and is still open for a wide range of parameters. In this paper, we study the degree versions of both theorems.More precisely, we give degree and t-degree versions of the Erdős–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 Erdős Matching Conjecture holds.