A semidefinite programming approach to a cross-intersection problem with measures

TL;DR

This paper introduces a semidefinite programming method to bound measures of cross-independent pairs in bipartite graphs, extending Hoffman's ratio bound, and applies it to solve a measure-based generalization of the Erdős-Ko-Rado theorem for cross-intersecting families.

Contribution

The paper develops a novel semidefinite programming approach to bound measures in bipartite graphs and applies it to a generalized measure version of the Erdős-Ko-Rado theorem.

Findings

Bound measures of cross-independent pairs using SDP

Extended Hoffman's ratio bound to measure settings

Solved a generalized measure Erdős-Ko-Rado problem

Abstract

We present a semidefinite programming approach to bound the measures of cross-independent pairs in a bipartite graph. This can be viewed as a far-reaching extension of Hoffman's ratio bound on the independence number of a graph. As an application, we solve a problem on the maximum measures of cross-intersecting families of subsets with two different product measures, which is a generalized measure version of the Erd\H{o}s-Ko-Rado theorem for cross-intersecting families with different uniformities.