
Veselin Jungić

Julian Sahasrabudhe
Keywords:
Pattern Avoidance, Abelian Groups, Arithmetic Progressions, Ramsey Theory
Abstract
Given a linear form $C_1X_1 + \cdots + C_nX_n$, 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 $(\alpha_1, \ldots , \alpha_n) \in G^n$ (with the $\alpha_i$ not all equal) to the equation $C_1X_1 + \cdots + C_nX_n = 0 $ to nonsolutions. This generalises a result of Hegarty about permutations of an abelian group avoiding arithmetic progressions. We also study the finite version of the problem suggested by Hegarty. We show that the number of permutations of $\mathbb{Z}/p\mathbb{Z}$ that map all 4term arithmetic progressions to nonprogressions, is asymptotically $e^{1}p!$.
Author Biographies
Veselin Jungić, Simon Fraser University
Department of Mathematics
Simon Fraser University
Burnaby, Canada
Julian Sahasrabudhe, Simon Fraser University
Department of Mathematics
Simon Fraser University
Burnaby, Canada