Séminaire Lotharingien de Combinatoire, 95B.102 (2026), 12 pp.

Nathan Lindzey

An Eventown Result for Permutations

Abstract. A family of permutations F ⊆ S_n is even-cycle-intersecting if σπ^-1 has an even cycle for all distinct σ,π ∈ F. We show that if F ⊆ S_n is an even-cycle-intersecting family of permutations, then |F| ≤ 2^n-1, and that the bound is met with equality when n is a power of 2 and F is a double-translate of a Sylow 2-subgroup of the symmetric group. This result can be seen as an analogue of the classical "eventown problem" for subsets, and it also answers a conjecture of János Körner on reversing families of permutations.

For the full version of this extended abstract, see arXiv:2601.12613.

Received: November 15, 2025. Accepted: February 15, 2026. Final version: April 1, 2026.

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