Families of Permutations with Forbidden Intersection-sizes

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Conjecture: for fixed t and n sufficiently large, the largest t-intersecting family of permutations has size (n−t)! and equality is achieved by t-cosets (fixing t points). Proof strategy uses weighted Hoffman via Delsarte and heavy use of representation theory of S_n and Young diagrams to compute eigenvalues of pseudo-adjacency matrices built as linear combinations of conjugacy-class Cayley graphs. The method succeeds for n large (gives stability results) but needs new ideas to reach the conjectured n ≥ 2t range.

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1. 0:00Start
2. 18:25Hoffman's Theorem
3. 25:58Adjacency Matrix
4. 43:20Open Problems
5. 51:17Spectral Proof of the Equality of the Equality Theorem
6. 52:08The Hungarian Matching Algorithm