# Kneser graphs are like Swiss cheese

## TL;DR

This paper investigates the structure of independent sets in Kneser and product graphs, demonstrating that small edge-spanning sets can be made independent by removing few vertices, thus extending previous results and revealing near-structural properties.

## Contribution

It introduces new bounds and structural insights for independent sets in Kneser graphs with fixed ratio parameters, strengthening prior theorems and connecting to graph removal techniques.

## Key findings

- Small edge-spanning sets can be made independent by removing few vertices.
- Any independent set is close to one depending on few coordinates.
- Results extend and strengthen previous theorems on Kneser graphs.

## Abstract

We prove that for a large family of product graphs, and for Kneser graphs $K(n,\alpha n)$ with fixed $\alpha <1/2$, the following holds. Any set of vertices that spans a small proportion of the edges in the graph can be made independent by removing a small proportion of the vertices of the graph. This allows us to strengthen the results of [DinurFR06] and [DinurF09], and show that any independent set in these graphs is almost contained in an independent set which depends on few coordinates. Our proof is inspired by, and follows some of the main ideas of, Fox's proof of the graph removal lemma [Fox11].