TY - JOUR

T1 - On symmetric intersecting families

AU - Ellis, David

AU - Kalai, Gil

AU - Narayanan, Bhargav

N1 - Funding Information:
The second author wishes to acknowledge support from ERC Advanced Grant 320924 , NSF, USA grant DMS-1300120 and BSF, Israel grant 2014290 , and the third author was partially supported by NSF, USA Grant DMS-1800521 . We would also like to thank two anonymous referees for their careful reading of the paper.

PY - 2020/5

Y1 - 2020/5

N2 - We make some progress on a question of Babai from the 1970s, namely: for n,k∈N with k≤n∕2, what is the largest possible cardinality s(n,k) of an intersecting family of k-element subsets of {1,2,…,n} admitting a transitive group of automorphisms? We give upper and lower bounds for s(n,k), and show in particular that [Formula presented] as n→∞ if and only if k=n∕2−ω(n)(n∕logn) for some function ω(⋅) that increases without bound, thereby determining the threshold at which ‘symmetric’ intersecting families are negligibly small compared to the maximum-sized intersecting families. We also exhibit connections to some basic questions in group theory and additive number theory, and pose a number of problems.

AB - We make some progress on a question of Babai from the 1970s, namely: for n,k∈N with k≤n∕2, what is the largest possible cardinality s(n,k) of an intersecting family of k-element subsets of {1,2,…,n} admitting a transitive group of automorphisms? We give upper and lower bounds for s(n,k), and show in particular that [Formula presented] as n→∞ if and only if k=n∕2−ω(n)(n∕logn) for some function ω(⋅) that increases without bound, thereby determining the threshold at which ‘symmetric’ intersecting families are negligibly small compared to the maximum-sized intersecting families. We also exhibit connections to some basic questions in group theory and additive number theory, and pose a number of problems.

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U2 - 10.1016/j.ejc.2020.103094

DO - 10.1016/j.ejc.2020.103094

M3 - Article

AN - SCOPUS:85080897037

VL - 86

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

SN - 0195-6698

M1 - 103094

ER -