Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, February 17, 2015, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Marcelo Gauy

Erdős-Ko-Rado in Random Families

The classical Erdős-Ko-Rado Theorem answers the following question: given a set A with n elements, what is the largest size of a family of subsets of size k(k-subsets) such that every two of them intersect? If n is at least 2k, the answer is found to be the binomial number n-1 choose k-1. We study a probabilistic version of this problem: given a random family of all k-subsets of A, where each k-subset is kept with probability p, what is the largest size of a family of k-subsets, among the chosen ones, such that every two of them intersect? This is a random variable and we determine its behaviour for almost all functions k and p of n. For k asymptotically larger than $n^{1/2+\epsilon}$, our results imply that this random variable goes through three phases. In the first phase it has size almost equal to the number of k subsets of our random set. In the third phase, its size is what is expected from the Erdős-Ko-Rado Theorem: around p times the binomial number n-1 choose k-1. The intermediary phase represents the transition between the two regimes and the random variable almost does not grow as p increases. A similar version of this problem was first studied by Balogh, Bohman and Mubayi in 2008. In recent years, many different groups studied related versions of this problem. This is joint work with Hiêp Hàn and Igor Oliveira.

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