We prove that for n sufficiently large, if A is a family of permutations
of {1,2,β¦,n} with no two permutations in A agreeing
exactly once, then β£Aβ£β€(nβ2)!, with equality holding only if
A is a coset of the stabilizer of 2 points. We also obtain a
Hilton-Milner type result, namely that if 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 $\{\sigma \in S_{n}:\ \sigma(1)=1,\sigma(2)=2,\
\#\{\textrm{fixed points of}\sigma \geq 5\} \neq 1\} \cup \{(1\ 3)(2\ 4),(1\
4)(2\ 3),(1\ 3\ 2\ 4),(1\ 4\ 2\ 3)\}.Weconjecturethatfort \in \mathbb{N},andfornsufficientlylargedependingont,if\mathcal{A}isfamilyofpermutationsof\{1,2,\ldots,n\}withnotwopermutationsin\mathcal{A}agreeingexactlyt-1times,then|\mathcal{A}| \leq (n-t)!,withequalityholdingonlyif\mathcal{A}isacosetofthestabilizeroftpoints.ThiscanbeseenasapermutationanalogueofaconjectureofErdoΛsonfamiliesofk$-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