Colorful fractional Helly theorem via weak saturation

TL;DR

This paper provides new combinatorial proofs for the fractional and colorful Helly theorems by reducing them to weak saturation problems, avoiding the use of exterior algebra and establishing optimal bounds.

Contribution

It introduces a novel combinatorial reduction of fractional Helly theorems to weak saturation, offering simpler proofs and new insights into these classical results.

Findings

Reduced fractional Helly theorem to weak saturation problem

Established optimal bounds for fractional and colorful Helly theorems

Provided shorter proofs without exterior algebra

Abstract

Two celebrated extensions of the classical Helly's theorem are the fractional Helly theorem and the colorful Helly theorem. Bulavka, Goodarzi, and Tancer recently established the optimal bound for the unified generalization of the fractional and the colorful Helly theorems using a colored extension of the exterior algebra. In this paper, we combinatorially reduce both the fractional Helly theorem and its colorful version to a classical problem in extremal combinatorics known as {weak saturation}. No such results connecting the fractional Helly theorem and weak saturation are known in the long history of literature. These reductions, along with basic linear algebraic arguments for the reduced weak saturation problems, let us give new short proofs of the optimal bounds for both the fractional Helly theorem and its colorful version without using exterior algebra.