Cross-Intersecting Families of Partial Permutations

Abstract

For positive integers r and n with r ≤ n, let Pn,r be the family of all sets {(x 1 , y 1 ), . . . , (xr, yr)} such that x 1 , . . . , xr are distinct elements of [n] := {1, . . . , n} and y 1 , . . . , yr are also distinct elements of [n]. Pn,n describes permutations of [n]. For r < n, Pn,r describes r-partial permutations of [n]. Families A 1 , . . . , A k of sets are said to be cross-intersecting if, for any distinct i and j in [k], any set in A i intersects any set in A j . A sharp bound for the sum of sizes of cross-intersecting subfamilies of Pn,n has recently been established by the author. We generalize this bound by showing that, if A 1 , . . . , A k are cross-intersecting subfamilies of Pn,r, thenWe also determine the structures for which the bound is attained when r < n. Our main tool is an extension of Katona's cyclic permutation method.

New Bounds on Even Cycle Creating Hamiltonian Paths Using Expander Graphs

“…In the proof of Theorem 1.5, we employed Lemma 3.1 to establish the inequality (2) H n (C 4 ) ≤ n!/2 H n (C 4 ) , and then we used Hamiltonian paths of a C 4 -free graph to get an asymptotically optimal lower bound on H n (C 4 ). In the following subsection we argue that using G-free graphs to prove lower bounds on H n (G) does not always yield asymptotically optimal results.…”

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