# Research

### An EKR theorem for matchings in the complete graph

The talk concerns the following higher-order analog of the Erdős-Ko-Rado theorem. For positive integers  and  with , let  be the family of all matchings of size in the complete graph . For any edge ee in , the family , which consists of all sets in  containing , is called the star centered at . We prove that if  and  is an intersecting family of matchings in , then , where  is an edge in . We also prove that equality holds if and only if  is a star. The main technique we use to prove the theorem is an analog of Katona’s elegant cycle method.