TY - CHAP

T1 - Maximal k-Wise ℓ -Divisible Set Families Are Atomic

AU - Gishboliner, Lior

AU - Sudakov, Benny

AU - Tomon, István

N1 - Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - 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 / ℓ ⌋. Yet, for infinitely many ℓ, Frankl and Odlyzko constructed families F as above of much bigger size 2Ω ( n log ℓ / ℓ ). 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 / ℓ. We completely resolve this old conjecture in a strong form, showing that | F| ≤ 2⌊ n / ℓ ⌋+ O(1 ) holds if k is chosen appropriately.

AB - 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 / ℓ ⌋. Yet, for infinitely many ℓ, Frankl and Odlyzko constructed families F as above of much bigger size 2Ω ( n log ℓ / ℓ ). 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 / ℓ. We completely resolve this old conjecture in a strong form, showing that | F| ≤ 2⌊ n / ℓ ⌋+ O(1 ) holds if k is chosen appropriately.

KW - Extremal combinatorics

KW - Intersections

KW - Set systems

UR - http://www.scopus.com/inward/record.url?scp=85114093242&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-83823-2_58

DO - 10.1007/978-3-030-83823-2_58

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AN - SCOPUS:85114093242

T3 - Trends in Mathematics

SP - 373

EP - 377

BT - Trends in Mathematics

PB - Springer Science and Business Media Deutschland GmbH

ER -