Abstract
For natural numbers$n,r\in \mathbb{N}$with$n\geqslant r$, the Kneser graph$K(n,r)$is the graph on the family of$r$-element subsets of$\{1,\ldots ,n\}$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 shall answer this question affirmatively as long as$r/n$is bounded away from$1/2$, even when the probability of retaining an edge of the Kneser graph is quite small. This gives us a random analogue of the Erdős–Ko–Rado theorem, since an independent set in the Kneser graph is the same as a uniform intersecting family. To prove our main result, we give some new estimates for the number of disjoint pairs in a family in terms of its distance from an intersecting family; these might be of independent interest.
Highlights
Over the past 20 years, a great deal of work has gone into proving ‘sparse random’ analogues of classical extremal results in combinatorics
We shall be interested in proving such a transference result for a central result in extremal set theory, the Erdos–Ko–Rado theorem
Lemma 3.1 follows by applying Proposition 3.2 to GA, the subgraph of the Kneser graph K (n, r ) induced by A
Summary
Over the past 20 years, a great deal of work has gone into proving ‘sparse random’ analogues of classical extremal results in combinatorics. Bohman and Mubayi [2], and, more recently, Hamm and Kahn [14], have obtained results which show that, under certain reasonable conditions, with high probability, the largest intersecting subfamily of a randomly chosen uniform family is trivial.
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