Permutations Destroying Arithmetic Progressions in Finite Cyclic Groups

Permutations Destroying Arithmetic Progressions in Finite Cyclic Groups

A permutation π of an abelian group G is said to destroy arithmetic progressions (APs) if, whenever (a, b, c) is a non-trivial 3-term AP in G, that is c − b = b − a and a, b, c are not all equal, then (π(a), π(b), π(c)) is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of Zn, for all n 6∈ {2, 3, 5, 7}. Here we prove, as a special case of a more gene...

Permutations Destroying Arithmetic Structure

Given a linear form C1X1 + · · · + CnXn, with coefficients in the integers, we characterize exactly the countably infinite abelian groups G for which there exists a permutation f that maps all solutions (α1, . . . , αn) ∈ Gn (with the αi not all equal) to the equation C1X1+ · · ·+CnXn = 0 to non-solutions. This generalises a result of Hegarty about permutations of an abelian group avoiding arit...

Rainbow Arithmetic Progressions in Finite Abelian Groups

For positive integers n and k, the anti-van der Waerden number of Zn, denoted by aw(Zn, k), is the minimum number of colors needed to color the elements of the cyclic group of order n and guarantee there is a rainbow arithmetic progression of length k. Butler et al. showed a reduction formula for aw(Zn, 3) = 3 in terms of the prime divisors of n. In this paper, we analagously define the anti-va...

Permutations Avoiding Arithmetic Progressions

Permutations that Destroy Arithmetic Progressions in Elementary p-Groups

Given an abelian group G, it is natural to ask whether there exists a permutation π of G that “destroys” all nontrivial 3-term arithmetic progressions (APs), in the sense that π(b)− π(a) 6= π(c)− π(b) for every ordered triple (a, b, c) ∈ G3 satisfying b − a = c − b 6= 0. This question was resolved for infinite groups G by Hegarty, who showed that there exists an AP-destroying permutation of G i...