The Erdős-Ko-Rado Theorem for 2-Pointwise and 2-Setwise Intersecting Permutations

In this paper we consider the conjectured Erdős-Ko-Rado property for $2$-pointwise and $2$-setwise intersecting permutations. Two permutations $\sigma,\tau \in \operatorname{Sym}(n)$ are $t$-setwise intersecting if there exists a $t$-subset $S$ of $\{1,2,\dots,n\}$ such that $S^\sigma = S^\tau$. Further, two permutations $\sigma,\tau \in \operatorname{Sym}(n)$ are $t$-pointwise intersecting if there exists a $t$-subset $S$ of $\{1,2,\dots,n\}$ such that $s^\sigma = s^\tau$ for each $s \in S$. A family of permutations $\mathcal{F} \subset \operatorname{Sym}(n)$ is called $t$-setwise (resp. $t$-pointwise) intersecting, if any two permutations in $\mathcal{F}$ are $t$-setwise (resp. $t$-pointwise) intersecting. We say that $\operatorname{Sym}(n)$ has the $t$-setwise intersecting property if for any family $\mathcal{F}$ of $t$-setwise intersecting permutations, $|\mathcal{F}| \leqslant t!(n-t)!$. Similarly, $\operatorname{Sym}(n)$ has the $t$-pointwise intersecting property if for any family $\mathcal{F}$ of $t$-pointwise intersecting permutations, $|\mathcal{F}| \leqslant (n-t)!$.Ellis ([``"Setwise intersecting families of permutations". J. Combin. Theory Ser. A, 119(4):825-849, 2012]), proved that if $n$ is sufficiently large relative to $t$, then $\operatorname{Sym}(n)$ has the $t$-setwise intersecting property. Ellis also conjectured that this result holds for all $n \geqslant t$. Ellis, Friedgut and Pilpel ["``Intersecting families of permutations." J. Amer. Math. Soc. 24(3):649-682, 2011] also proved that for $n$ sufficiently large relative to $t$, $\operatorname{Sym}(n)$ has the $t$-pointwise intersecting property. It is also conjectured that $\operatorname{Sym}(n)$ has the $t$-pointwise intersecting property for $n\geqslant 2t+1$. In this work, we prove these two conjectures for $\operatorname{Sym}(n)$ when $t=2$.

Almost Intersecting Families

Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of $[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \varnothing$ for all $F, F' \in \mathcal F$. It is called almost intersecting if it is not intersecting but to every $F \in \mathcal F$ there is at most one $F'\in \mathcal F$ satisfying $F \cap F' = \varnothing$. Gerbner et al. proved that if $n \geq 2k + 2$ then $|\mathcal F| \leqslant {n - 1\choose k - 1}$ holds for almost intersecting families. Our main result implies the considerably stronger and best possible bound $|\mathcal F| \leqslant {n - 1\choose k - 1} - {n - k - 1\choose k - 1} + 2$ for $n > (2 + o(1))k$, $k\ge 3$.

On symmetric intersecting families of vectors

A family of vectors in [k] n is said to be intersecting if any two of its elements agree on at least one coordinate. We prove, for fixed k ≥ 3, that the size of any intersecting subfamily of [k] n invariant under a transitive group of symmetries is o(k n ), which is in stark contrast to the case of the Boolean hypercube (where k = 2). Our main contribution addresses limitations of existing technology: while there are now methods, first appearing in work of Ellis and the third author, for using spectral machinery to tackle problems in extremal set theory involving symmetry, this machinery relies crucially on the interplay between up-sets, biased product measures, and threshold behaviour in the Boolean hypercube, features that are notably absent in the problem considered here. To circumvent these barriers, introducing ideas that seem of independent interest, we develop a variant of the sharp threshold machinery that applies at the level of products of posets.