Removal and Stability for Erdős-Ko-Rado [article]

Shagnik Das, Tuan Tran
2016 arXiv   pre-print
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 Erdős-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 γ n < k < (12 - γ)n, a set family with few disjoint pairs
more » ... can be made intersecting by removing few sets. We provide a simple proof of a removal lemma for large families, showing that families of size close to ℓn-1k-1 with relatively few disjoint pairs must be close to a union of ℓ stars. Our lemma holds for a wide range of uniformities; in particular, when ℓ = 1, the result holds for all 2 < k < n/2 and provides sharp quantitative estimates. We use this removal lemma to settle a question of Bollobás, Narayanan and Raigorodskii regarding the independence number of random subgraphs of the Kneser graph K(n,k). The Erdős-Ko-Rado theorem shows α(K(n,k)) = n-1k-1. For some constant c > 0 and k < cn, we determine the sharp threshold for when this equality holds for random subgraphs of K(n,k), and provide strong bounds on the critical probability for k <12 (n-3).
arXiv:1412.7885v3 fatcat:vzugstetpjf3tfnrdlzlyt7mpu