Constructions and nonexistence results for suitable sets of permutations

TL;DR

This paper investigates the existence and nonexistence of suitable permutation sets with specific ordering properties, providing new constructions, relations between parameters, and both exact and asymptotic nonexistence results.

Contribution

It introduces new examples of suitable permutation sets, relates different parameter cases, and proves both exact and asymptotic nonexistence theorems.

Findings

New suitable permutation sets for specific parameters

A construction method linking different parameter cases

Exact nonexistence proof using combinatorial arguments

Abstract

A set of $N$ permutations of $\{1,2,\dots,v\}$ is $(N,v,t)$-suitable if each symbol precedes each subset of $t-1$ others in at least one permutation. The central problems are to determine the smallest $N$ for which such a set exists for given $v$ and $t$, and to determine the largest $v$ for which such a set exists for given $N$ and $t$. These extremal problems were the subject of classical studies by Dushnik in 1950 and Spencer in 1971. We give examples of suitable sets of permutations for new parameter triples $(N,v,t)$. We relate certain suitable sets of permutations with parameter $t$ to others with parameter $t+1$, thereby showing that one of the two infinite families recently presented by Colbourn can be constructed directly from the other. We prove an exact nonexistence result for suitable sets of permutations using elementary combinatorial arguments. We then establish an asymptotic nonexistence result using Ramsey's theorem.