# A quantitative Lov\'asz criterion for Property B

## TL;DR

This paper provides an exact quantitative criterion based on edge intersections for determining non-2-colorability in hypergraphs, extending Lovász's classic criterion and characterizing extremal cases.

## Contribution

It introduces a precise quantitative version of Lovász's criterion for hypergraph 2-colorability and characterizes all extremal hypergraphs achieving this bound.

## Key findings

- Established an exact count of intersecting edge pairs for non-2-colorable hypergraphs.
- Characterized all extremal hypergraphs meeting the criterion.
- Combined combinatorial and probabilistic methods in the proof.

## Abstract

A well known observation of Lov\'asz is that if a hypergraph is not $2$-colorable, then at least one pair of its edges intersect at a single vertex. %This very simple criterion turned out to be extremly useful . In this short paper we consider the quantitative version of Lov\'asz's criterion. That is, we ask how many pairs of edges intersecting at a single vertex, should belong to a non $2$-colorable $n$-uniform hypergraph? Our main result is an {\em exact} answer to this question, which further characterizes all the extremal hypergraphs. The proof combines Bollob\'as's two families theorem with Pluhar's randomized coloring algorithm.