Maximal k-Wise ℓ -Divisible Set Families Are Atomic

Lior Gishboliner, Benny Sudakov, István Tomon*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


Let F⊂ 2[ n ] such that the intersection of any two members of F has size divisible by ℓ. By the famous Eventown theorem, if ℓ= 2 then | F| ≤ 2 n / 2 , and this bound can be achieved by ‘atomic’ construction, i.e. splitting the ground set into disjoint pairs and taking their arbitrary unions. Similarly, splitting the ground set into disjoint sets of size ℓ gives a family with pairwise intersections divisible by ℓ and size 2 n / . Yet, for infinitely many ℓ, Frankl and Odlyzko constructed families F as above of much bigger size 2Ω ( n log / ). On the other hand, in 1983 they conjectured that for every ℓ there exists some k such that if any k distinct members of F have an intersection of size divisible by ℓ, then | F| ≤ 2( 1 + o ( 1 ) ) n / . We completely resolve this old conjecture in a strong form, showing that | F| ≤ 2 n / + O(1 ) holds if k is chosen appropriately.

Original languageEnglish
Title of host publicationTrends in Mathematics
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages5
StatePublished - 2021
Externally publishedYes

Publication series

NameTrends in Mathematics
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X


  • Extremal combinatorics
  • Intersections
  • Set systems


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