On Reverse-Free Codes and Permutations

Abstract

A set F of ordered k-tuples of distinct elements of an n-set is pairwise reverse free if it does not contain two ordered k-tuples with the same pair of elements in the same pair of coordinates in reverse order. Let F (n, k) be the maximum size of a pairwise reverse-free set. In this paper we focus on the case of 3-tuples and prove lim F (n, 3)/ n 3 = 5/4, more exactly, 5 24 n 3 − 1 2 n 2 − O(n log n) < F (n, 3) ≤ 5 24 n 3 − 1 2 n 2 + 5 8 n, and here equality holds when n is a power of 3. Many problems remain open.

Path Separation by Short Cycles

“…Cibulka has proved, improving an earlier bound of Füredi et al. that (A family satisfying the same condition is called full of flips in .) Our present interest in this question is motivated by the following observation.…”

“…(A family satisfying the same condition is called full of flips in [5].) Our present interest in this question is motivated by the following observation.…”

Setwise intersecting families of permutations

“…SinceÑ is invertible, there is a unique choice of the u α 's satisfying (9). However, there are two problems: we cannot choose u to be supported only on even permutations (or only on odd permutations), and since |X α | ≤ n!/(n − t + 1) for each α ∈ F n,t−1 , we cannot demand that max{|u α | : α ∈ F n,t−1 } ≤ O t (1/n!).…”

Optimal Multivalued Shattering

“…We define forb(n, F ) = max{|A| : A is a simple 0-1 matrix without configuration F of n columns}. (9) A simple (0,1)-matrix A naturally corresponds to a set system F A taking the rows as characteristic In [7] it was proved that…”