August 6, 2014
Delete the edges of a Kneser graph independently of each other with some probability: for what probabilities is the independence number of this random graph equal to the independence number of the Kneser graph itself? We prove a sharp threshold result for this question in certain regimes. Since an independent set in the Kneser graph is the same as a uniform intersecting family, this gives us a random analogue of the Erd\H{o}s-Ko-Rado theorem.
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September 30, 2015
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 oth...
September 5, 2016
For natural numbers $n,r \in \mathbb{N}$ with $n\ge r$, the Kneser graph $K(n,r)$ is the graph on the family of $r$-element subsets of $\{1,\dots,n\}$ in which two sets are adjacent if and only if they are disjoint. Delete the edges of $K(n,r)$ with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We answer this question affirmatively as long as $r/n$ is bounded away fro...
February 19, 2015
Denote by $K_p(n,k)$ the random subgraph of the usual Kneser graph $K(n,k)$ in which edges appear independently, each with probability $p$. Answering a question of Bollob\'as, Narayanan, and Raigorodskii,we show that there is a fixed $p<1$ such that a.s. (i.e., with probability tending to 1 as $k \to \infty$) the maximum independent sets of $K_p(2k+1, k)$ are precisely the sets $\{A\in V(K(2k+1,k)): x\in A\}$ ($x\in [2k+1]$). We also complete the determination of the order ...
May 6, 2021
For positive integers $n$ and $k$ with $n\geq 2k+1$, the Kneser graph $K(n,k)$ is the graph with vertex set consisting of all $k$-sets of $\{1,\dots,n\}$, where two $k$-sets are adjacent exactly when they are disjoint. The independent sets of $K(n,k)$ are $k$-uniform intersecting families, and hence the maximum size independent sets are given by the Erd\H{o}s-Ko-Rado Theorem. Let $K_p(n,k)$ be a random spanning subgraph of $K(n,k)$ where each edge is included independently wi...
December 26, 2014
A $k$-uniform family of subsets of $[n]$ is intersecting if it does not contain a disjoint pair of sets. The study of intersecting families is central to extremal set theory, dating back to the seminal Erd\H{o}s-Ko-Rado theorem of 1961 that bounds the size of the largest such families. A recent trend has been to investigate the structure of set families with few disjoint pairs. Friedgut and Regev proved a general removal lemma, showing that when $\gamma n \le k \le (\tfrac12 ...
February 27, 2018
The Kneser hypergraph ${\rm KG}^r_{n,k}$ is an $r$-uniform hypergraph with vertex set consisting of all $k$-subsets of $\{1,\ldots,n\}$ and any collection of $r$ vertices forms an edge if their corresponding $k$-sets are pairwise disjoint. The random Kneser hypergraph ${\rm KG}^r_{n,k}(p)$ is a spanning subhypergraph of ${\rm KG}^r_{n,k}$ in which each edge of ${\rm KG}^r_{n,k}$ is retained independently of each other with probability $p$. The independence number of random su...
February 22, 2009
In this note, we investigate some properties of local Kneser graphs defined in [8]. In this regard, as a generalization of the Erd${\rm \ddot{o}}$s-Ko-Rado theorem, we characterize the maximum independent sets of local Kneser graphs. Next, we present an upper bound for their chromatic number.
April 18, 2020
In this paper, we study the maximum degree in non-empty induced subgraphs of the Kneser graph $KG(n,k)$. One of the main results asserts that, for $k>k_0$ and $n>64k^2$, whenever a non-empty subgraph has $m\ge k{n-2\choose k-2}$ vertices, its maximum degree is at least $\frac 12(1-\frac {k^2}n) m - {n-2\choose k-2}\ge 0.49 m$. This bound is essentially best possible. One of the intermediate steps is to obtain structural results on non-empty subgraphs with small maximum degree...
September 12, 2014
We investigate the asymptotic version of the Erd\H{o}s-Ko-Rado theorem for the random $k$-uniform hypergraph $\mathcal{H}^k(n,p)$. For $2 \leq k(n) \leq n/2$, let $N=\binom{n}k$ and $D=\binom{n-k}k$. We show that with probability tending to 1 as $n\to\infty$, the largest intersecting subhypergraph of $\mathcal{H}^k(n,p)$ has size $(1+o(1))p\frac kn N$, for any $p\gg \frac nk\ln^2\!\left(\frac nk\right)D^{-1}$. This lower bound on $p$ is asymptotically best possible for $k=\Th...
February 3, 2015
A Kneser graph $KG_{n,k}$ is a graph whose vertices are in one-to-one correspondence with $k$-element subsets of $[n],$ with two vertices connected if and only if the corresponding sets do not intersect. A famous result due to Lov\'asz states that the chromatic number of a Kneser graph $KG_{n,k}$ is equal to $n-2k+2$. In this paper we study the chromatic number of a random subgraph of a Kneser graph $KG_{n,k}$ as $n$ grows. A random subgraph $KG_{n,k}(p)$ is obtained by inclu...