On the stability of some Erd\H{o}s--Ko--Rado type results

TL;DR

This paper investigates the stability of the independence number in certain random subgraphs related to Erd ext{o}s--Ko--Rado type graphs, proving that large independent sets maintain a specific structure with high probability.

Contribution

It establishes the stability of the independence number for random subgraphs of $G(n,r,1)$ when $r$ is constant and $p=1/2$, revealing structural properties of large independent sets.

Findings

Large independent sets have a predefined structure

Stability holds for constant r and p=1/2

Independence number remains unchanged with high probability

Abstract

Consider classical Kneser's graph $K(n,r)$: for two natural numbers $ r, n $ such that $r \le n / 2$, its vertices are all the subsets of $[n]=\{1,2,\ldots,n\}$ of size $r$, and two such vertices are adjacent if the corresponding subsets are disjoint. The Erd\H{o}s--Ko--Rado theorem states that the size of the largest independent set in this graph is $\binom{n-1}{r-1}$. Now let us delete each edge of the graph $K(n,r)$ with some fixed probability $p$ independently of each other. Quite surprisingly, the independence number of such random subgraph $K_p(n,r)$ of the graph $ K(n,r) $ is, with high probability, the same as the independence number of the initial graph. This phenomenon is called the stability of the independence number. This paper concerns the independence number of random subgraphs of the graph $G(n,r,1)$, which vertices are the same as in the Kneser graph and there is an edge between two vertices if they intersect in exactly one element. We prove the stability result for constant $r$ and $p=\frac{1}{2}$. To do that, we show that any sufficiently large independent set in $G(n,r,1)$ has some predefined structure, and this is a result of independent interest.