Forbidding just one intersection, for permutations

We prove that for $n$ sufficiently large, if $A$ is a family of permutations of $\{1,2,\ldots,n\}$ with no two permutations in $\mathcal{A}$ agreeing exactly once, then $|\mathcal{A}| \leq (n-2)!$, with equality holding only if $\mathcal{A}$ is a coset of the stabilizer of 2 points. We also obtain a Hilton-Milner type result, namely that if $\mathcal{A}$ is such a family which is not contained within a coset of the stabilizer of 2 points, then it is no larger than the family $\{\sigma \in S_{n}:\ \sigma(1)=1,\sigma(2)=2,\ \#\{\textrm{fixed points of}\sigma \geq 5\} \neq 1\} \cup \{(1\ 3)(2\ 4),(1\ 4)(2\ 3),(1\ 3\ 2\ 4),(1\ 4\ 2\ 3)\}$. We conjecture that for$t \in \mathbb{N}$, and for$n$sufficiently large depending on$t$, if$\mathcal{A}$is family of permutations of$\{1,2,\ldots,n\}$with no two permutations in$\mathcal{A}$agreeing exactly$t-1$times, then$|\mathcal{A}| \leq (n-t)!$, with equality holding only if$\mathcal{A}$is a coset of the stabilizer of$t$points. This can be seen as a permutation analogue of a conjecture of Erd\H{o}s on families of$k$-element sets with a forbidden intersection, proved by Frankl and F\"uredi in [P. Frankl and Z. F\"uredi, Forbidding Just One Intersection, Journal of Combinatorial Theory, Series A, Volume 39 (1985), pp. 160-176].Comment: 26 page

A quasi-stability result for dictatorships in SnS_{n}

We prove that Boolean functions on $S_{n}$ whose Fourier transform is highly concentrated on the first two irreducible representations of $S_n$, are close to being unions of cosets of point-stabilizers. We use this to give a natural proof of a stability result on intersecting families of permutations, originally conjectured by Cameron and Ku, and first proved by the first author. We also use it to prove a `quasi-stability' result for an edge-isoperimetric inequality in the transposition graph on $S_n$, namely that subsets of $S_n$ with small edge-boundary in the transposition graph are close to being unions of cosets of point-stabilizers.Comment: Introduction updated and expanded; 'Background' section expanded; references update

Low-degree Boolean functions on SnS_n, with an application to isoperimetry

We prove that Boolean functions on $S_n$, whose Fourier transform is highly concentrated on irreducible representations indexed by partitions of $n$ whose largest part has size at least $n-t$, are close to being unions of cosets of stabilizers of $t$-tuples. We also obtain an edge-isoperimetric inequality for the transposition graph on $S_n$ which is asymptotically sharp for subsets of $S_n$ of size $n!/\textrm{poly}(n)$, using eigenvalue techniques. We then combine these two results to obtain a sharp edge-isoperimetric inequality for subsets of $S_n$ of size $(n-t)!$, where $n$ is large compared to $t$, confirming a conjecture of Ben Efraim in these cases.Comment: Minor corrections to statements of Lemmas 15 and 16. A prior theorem, cited in the Intro. of the previous version (Theorem 2) has recently been found to be false. This does not affect the rest of the paper. We have amended the statement of Theorem 2 and provided a counterexample to the original statement. This counterexample shows that our main theorem (Theorem 3) is sharper than we first though