TY - JOUR

T1 - APPROXIMATION BY JUNTAS IN THE SYMMETRIC GROUP, AND FORBIDDEN INTERSECTION PROBLEMS

AU - Ellis, David

AU - Lifshitz, Noam

N1 - Publisher Copyright:
© 2022 Duke University Press. All rights reserved.

PY - 2022

Y1 - 2022

N2 - A family of permutations F ⊂ Sn is said to be t -intersecting if any two permutations in F agree on at least t points. It is said to be .t − 1/-intersection-free if no two permutations in F agree on exactly t − 1 points. If S; T ⊂ 11; 2;:::; nºwith jS j D jT j, and π W S → T is a bijection, then the π -star in Sn is the family of all permutations that agree with π on S . An s-star is a π-star such that π is a bijection between sets of size s. Friedgut and Pilpel, and independently the first author, showed that if F ⊂ Sn is t -intersecting, and n is sufficiently large depending on t , then jF j ≤ .n − t /Š; this proved a conjecture of Deza and Frankl from 1977. Here, we prove a considerable strengthening of the Deza–Frankl conjecture, namely, that if n is sufficiently large depending on t , and F ⊂ Sn is .t − 1/-intersection-free, then jF j ≤ .n − t /Š, with equality iff F is a t -star. The main ingredient of our proof is a “junta approximation” result, namely, that any .t − 1/-intersection-free family of permutations is essentially contained in a t - intersecting junta (a “junta” being a union of boundedly many O.1/-stars). The proof of our junta approximation result relies, in turn, on (i) a weak regularity lemma for families of permutations (which outputs a junta whose stars are intersected by F in a weakly pseudorandom way), (ii) a combinatorial argument that “bootstraps” the weak notion of pseudorandomness into a stronger one, and finally (iii) a spectral argument for highly pseudorandom fractional families. Our proof employs four different notions of pseudorandomness, three being combinatorial in nature and one being algebraic. The connection we demonstrate between these notions of pseudorandomness may find further applications.

AB - A family of permutations F ⊂ Sn is said to be t -intersecting if any two permutations in F agree on at least t points. It is said to be .t − 1/-intersection-free if no two permutations in F agree on exactly t − 1 points. If S; T ⊂ 11; 2;:::; nºwith jS j D jT j, and π W S → T is a bijection, then the π -star in Sn is the family of all permutations that agree with π on S . An s-star is a π-star such that π is a bijection between sets of size s. Friedgut and Pilpel, and independently the first author, showed that if F ⊂ Sn is t -intersecting, and n is sufficiently large depending on t , then jF j ≤ .n − t /Š; this proved a conjecture of Deza and Frankl from 1977. Here, we prove a considerable strengthening of the Deza–Frankl conjecture, namely, that if n is sufficiently large depending on t , and F ⊂ Sn is .t − 1/-intersection-free, then jF j ≤ .n − t /Š, with equality iff F is a t -star. The main ingredient of our proof is a “junta approximation” result, namely, that any .t − 1/-intersection-free family of permutations is essentially contained in a t - intersecting junta (a “junta” being a union of boundedly many O.1/-stars). The proof of our junta approximation result relies, in turn, on (i) a weak regularity lemma for families of permutations (which outputs a junta whose stars are intersected by F in a weakly pseudorandom way), (ii) a combinatorial argument that “bootstraps” the weak notion of pseudorandomness into a stronger one, and finally (iii) a spectral argument for highly pseudorandom fractional families. Our proof employs four different notions of pseudorandomness, three being combinatorial in nature and one being algebraic. The connection we demonstrate between these notions of pseudorandomness may find further applications.

UR - http://www.scopus.com/inward/record.url?scp=85132941893&partnerID=8YFLogxK

U2 - 10.1215/00127094-2021-0050

DO - 10.1215/00127094-2021-0050

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AN - SCOPUS:85132941893

SN - 0012-7094

VL - 171

SP - 1417

EP - 1467

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

IS - 7

ER -