On Reverse-Free Codes and Permutations

A set $\mathcal{F}$ 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 $\lim F(n,3)/\binom{n}{3}=5/4$, more exactly, $\frac{5}{24}n^3-\frac{1}{2}n^2-O(n\log n)<F(n,3)\leq\frac{5}{24}n^3-\frac{1}{2}n^2+\frac{5}{8}n$, and here equality holds when $n$ is a power of 3. Many problems remain open.

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