On symmetric intersecting families

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.Comment: Minor change to the statement (and proof) of Theorem 1.4; the authors thank Nathan Keller and Omri Marcus for pointing out a mistake in the previous versio

Disjoint induced subgraphs of the same order and size

For a graph $G$, let $f(G)$ be the largest integer $k$ for which there exist two vertex-disjoint induced subgraphs of $G$ each on $k$ vertices, both inducing the same number of edges. We prove that $f(G) \ge n/2 - o(n)$ for every graph $G$ on $n$ vertices. This answers a question of Caro and Yuster.Comment: 25 pages, improved presentation, fixed misprints, European Journal of Combinatoric