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An almost complete t-intersection theorem for permutations
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An almost complete t-intersection theorem for permutations
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For any $\epsilon>0$ and $n>(1+\epsilon)t$, $n>n_0(\epsilon)$ we determine the size of the largest $t$-intersecting family of permutations, as well as give a sharp stability result. This resolves a conjecture of Ellis, Friedgut and Pilpel (2011) and shows the validity of conjectures of Frankl and Deza (1977) and Cameron (1988) for $n>(1+\epsilon )t$. We note that, for this range of parameters, the extremal examples are not necessarily trivial, and that our statement is analogous to the celebrated Ahlswede-Khachatrian theorem. The proof is based on the refinement of the method of spread approximations, recently introduced by Kupavskii and Zakharov (2022).
Forward citations
Cited by 2 Pith papers
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A Complete Intersection Theorem for Large Permutation Groups
Proves that for sufficiently large n the maximum t-intersecting families in S_n are the fixed-point families F_{n,t,r}, resolving the Deza-Frankl problem asymptotically.
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A unified approach to cross-intersection problems with applications to Hilton--Milner type theorems and stability
A fingerprint/t-cover iteration determines extremal and stable cross t-intersecting k-uniform families for large n, including product EKR for spread systems and t-diversity bounds.
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