Abstract
The extremal problems regarding the maximum possible size of intersecting families of various combinatorial objects have been extensively studied. In this paper, we investigate supersaturation extensions, which in this context ask for the minimum number of disjoint pairs that must appear in families larger than the extremal threshold. We study the minimum number of disjoint pairs in families of permutations and in $k$-uniform set families, and determine the structure of the optimal families. Our main tool is a removal lemma for disjoint pairs. We also determine the typical structure of $k$-uniform set families without matchings of size $s$ when $n \ge 2sk + 38s^4$, and show that almost all $k$-uniform intersecting families on vertex set $[n]$ are trivial when $n\ge (2+o(1))k$.
Highlights
Determining the size of intersecting families of discrete objects is a line of research with a long history, originating in extremal set theory
One might expect a family of permutations with the minimum number of disjoint pairs the electronic journal of combinatorics 26(2) (2019), #P2.34 to contain large intersecting subfamilies, and a candidate construction is the union of an appropriate number of cosets
Following the the electronic journal of combinatorics 26(2) (2019), #P2.34 same process for the family T (n, t), we see that the number of disjoint pairs is decreased by exactly (k − 1)Dn−1 each time we remove a permutation from the last coset in T (n, t)
Summary
Determining the size of intersecting families of discrete objects is a line of research with a long history, originating in extremal set theory. One might expect a family of permutations with the minimum number of disjoint pairs the electronic journal of combinatorics 26(2) (2019), #P2.34 to contain large intersecting subfamilies, and a candidate construction is the union of an appropriate number of cosets. We consider the supersaturation extension of the original Erdos–Ko–Rado Theorem, where one seeks to minimise the number of disjoint pairs of sets in a k-uniform family of s subsets of [n]. That is, when n is sufficiently large and the families are of small size, the initial segments of the lexicographic order minimise the number of disjoint pairs, confirming the Bollobas–Leader conjecture in this range. Rather than considering the supersaturation phenomenon, we describe the typical structure of set families with a given property, showing that almost all such families are subfamilies of the trivial extremal constructions.
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