On regular 3-wise intersecting families

Abstract

Ellis and the third author showed, verifying a conjecture of Frankl, that any 3-wise intersecting family of subsets of {1, 2, . . . , n} admitting a transitive automorphism group has cardinality o(2 n ), while a construction of Frankl demonstrates that the same conclusion need not hold under the weaker constraint of being regular. Answering a question of Cameron, Frankl and Kantor from 1989, we show that the restriction of admitting a transitive automorphism group may be relaxed significantly: we prove that any 3-wise intersecting family of subsets of {1, 2, . . . , n} that is regular and increasing has cardinality o(2 n ).

On symmetric intersecting families of vectors

“…It is therefore natural to ask what happens when one further imposes a symmetry requirement on the intersecting family under consideration, the most natural such requirement being that the family admit a transitive group of automorphisms. Indeed, this direction was proposed by Babai [2] a few decades back, and has since been rather fruitful; see [9,5] for some classical results, and [7,6,13] for more recent developments.…”

“…While the statement of Theorem 1.1 is inherently attractive (in our opinion), there are deeper considerations which make the result particularly interesting from the point of view of technique, as we now explain. Ellis and the third author [7], in resolving a conjecture of Frankl [9] about symmetric 3-wise intersecting families, introduced some spectral machinery for tackling problems in extremal set theory involving symmetry; this framework has since been successfully adapted -see [6,13], for example -to resolve a few other old extremal problems in the Boolean hypercube involving symmetry constraints. This approach depends crucially on exploiting the interplay between up-sets, biased product measures, and 'sharp threshold' behaviour in the Boolean hypercube; these features are absent in the problem under consideration here, and indeed, our main contribution is a method to circumvent these barriers.…”

“…Concretely, working in [3] n with the natural partial order induced by point-wise comparison, compressing an intersecting family 'upwards' preserves the intersection condition but not the symmetry group, while alternately, replacing a family by its 'up-closure' in the natural partial order preserves the symmetry group but not the intersection condition; furthermore, there appears to be no natural analogue in [3] n of the p-biased product measures on [2] n that are at the heart of the arguments in [7,6,13].…”