Typical intersecting families at $n=2k+1$ and $n=2k+2$

Abstract

A family of sets is intersecting if every two members intersect, and trivial if all its members contain a common element. We determine the typical structure of $k$-uniform intersecting families on $2k+1$ and $2k+2$ elements as $k\to\infty$. For $n=2k+2$, we prove that almost all intersecting families are trivial and that their number is \[ (2k+2+o(1))\,2^{\binom{2k+1}{k-1}}. \] Together with Yang's recent result for $n\ge 2k+3$, this settles a conjecture of Balogh, Garcia, Li, and Wagner. For $n=2k+1$, almost all intersecting families are nontrivial. We prove that, as conjectured by the same authors, a typical intersecting family is close to a full star: its members outside the star form components of size at most two in the graph joining sets that intersect in $k-1$ elements. We also obtain an asymptotic formula for the number of intersecting families in this case, with an explicit second-order term in the exponent. Our proof combines Sapozhenko's graph container method and stability in Kneser graphs to control families far from every star, and a polymer model and cluster expansion to enumerate families close to a fixed star.