A family of permutations $\mathcal{F} \subseteq S_n$ is even-cycle-intersecting if $σπ^{-1}$ has an even cycle for all $σ,π\in \mathcal{F}$. We show that if $\mathcal{F} \subseteq S_n$ is an even-cycle-intersecting family of permutations, then $|\mathcal{F}| \leq 2^{n-1}$, and that equality holds when $n$ is a power of 2 and $\mathcal{F}$ is a double-translate of a Sylow 2-subgroup of $S_n$. This result can be seen as an analogue of the classical eventown problem for subsets and it confirms a conjecture of János Körner on maximum reversing families of the symmetric group. Along the way, we show that the canonically intersecting families of $S_n$ are also the extremal odd-cycle-intersecting families of $S_n$ for all even $n$. While the latter result has less combinatorial significance, its proof uses an interesting new character-theoretic identity that might be of independent interest in algebraic combinatorics.

翻译：若对任意 $\sigma,\pi\in \mathcal{F}$，置换 $\sigma\pi^{-1}$ 均包含偶循环，则称置换族 $\mathcal{F} \subseteq S_n$ 为偶循环相交族。我们证明：若 $\mathcal{F} \subseteq S_n$ 是偶循环相交置换族，则 $|\mathcal{F}| \leq 2^{n-1}$；当 $n$ 为 2 的幂且 $\mathcal{F}$ 是 $S_n$ 的西洛 2-子群的双重平移时，等号成立。该结果可视为经典偶数镇问题在置换情形下的类比，并证实了 János Körner 关于对称群最大反转族的猜想。在此过程中，我们证明了对于所有偶数 $n$，$S_n$ 的典范相交族同时也是其极值奇循环相交族。尽管后一结果的组合意义较弱，但其证明使用了一个有趣的、可能对代数组合学具有独立价值的新特征恒等式。