Uniqueness for 2-intersecting families of permutations and perfect matchings

Gilad Chase, Neta Dafni, Yuval Filmus and Nathan Lindzey

We characterize the largest $2$-intersecting families of permutations of $\{1,2,\ldots,n\}$ and of perfect matchings of the complete graph $K_{2n}$ for all $n \geq 2$: the consist, respectively, of all permutations mapping $i_1$ to $j_1$ and $i_2 \neq i_1$ to $j_2 \neq j_1$, and of all perfect matchings containing two fixed edges.

The characterization uses techniques from our work Complexity measures on symmetric group and beyond, and answers open questions in Meagher and Razafimahatratra, 2-intersecting permutations and Fallat, Meagher and Shirazi, The Erdős–Ko–Rado theorem for 2-intersecting families of perfect matchings.


 title = {Uniqueness for 2-Intersecting Families of Permutations and Perfect Matchings},
 author = {Gilad Chase and Neta Dafni and Yuval Filmus and Nathan Lindzey},
 howpublished = {Manuscript},
 year = {2022}
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