Abstract
The celebrated Erdős-Ko-Rado theorem shows that for $n \ge 2k$ the largest intersecting $k$-uniform set family on $[n]$ has size $\binom{n-1}{k-1}$. It is natural to ask how far from intersecting larger set families must be. Katona, Katona and Katona introduced the notion of most probably intersecting families, which maximise the probability of random subfamilies being intersecting.We consider the most probably intersecting problem for $k$-uniform set families. We provide a rough structural characterisation of the most probably intersecting families and, for families of particular sizes, show that the initial segment of the lexicographic order is optimal.
Highlights
A family of sets F is said to be intersecting if F1 ∩ F2 = ∅ for all F1, F2 ∈ F
A central result in extremal set theory is the Erdos-Ko-Rado theorem, which determines the largest size of an intersecting k-uniform family over [n]
One may investigate the appearance of disjoint pairs in larger families of sets
Summary
A central result in extremal set theory is the Erdos-Ko-Rado theorem, which determines the largest size of an intersecting k-uniform family over [n]. Given this extremal result, one may investigate the appearance of disjoint pairs in larger families of sets. We provide an approximate structural result, and are able to determine the extremal hypergraphs exactly for some ranges of values of m. These mark the first general results for the probabilistic supersaturation problem for k-uniform set families. We discuss the history of the supersaturation problem for intersecting families, before introducing the probabilistic version of Katona, Katona and Katona and presenting our new results
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