# On symmetric intersecting families

## TL;DR

This paper investigates the maximum size of symmetric intersecting families of subsets, providing bounds and identifying when such families are asymptotically negligible compared to the largest possible, with connections to group theory and number theory.

## Contribution

It establishes bounds for symmetric intersecting families and determines the threshold where they become negligible, advancing understanding of symmetry constraints in combinatorial set families.

## Key findings

- s(n,k) = o (binom(n-1,k-1)) under certain conditions
- Identifies the threshold for when symmetric families are negligible
- Connects combinatorics with group theory and additive number theory

## Abstract

We make some progress on a question of Babai from the 1970s, namely: for $n, k \in \mathbb{N}$ with $k \le n/2$, what is the largest possible cardinality $s(n,k)$ of an intersecting family of $k$-element subsets of $\{1,2,\ldots,n\}$ admitting a transitive group of automorphisms? We give upper and lower bounds for $s(n,k)$, and show in particular that $s(n,k) = o (\binom{n-1}{k-1})$ as $n \to \infty$ if and only if $k = n/2 - \omega(n)(n/\log n)$ for some function $\omega(\cdot)$ 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.