On permutations avoiding arithmetic progressions

Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length 5 were constructed in (Davis et al, 1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length 7. We construct a permutation of the integers avoiding arithmetic progressions of length 6. We also prove a lower bound of 1 2 on the lower density of subsets of positive inte...

Permutations Avoiding Arithmetic Progressions

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...

Enumerating Permutations that Avoid Three Term Arithmetic Progressions

It is proved that the number of permutations of the set {1, 2, 3, . . . , n} that avoid three term arithmetic progressions is at most (2.7) n 21 for n ≥ 11 and at each end of any such permutation, at least ⌊ 2 ⌋−6 entries have the same parity.

Words avoiding repetitions in arithmetic progressions

Carpi constructed an infinite word over a 4-letter alphabet that avoids squares in all subsequences indexed by arithmetic progressions of odd difference. We show a connection between Carpi’s construction and the paperfolding words. We extend Carpi’s result by constructing uncountably many words that avoid squares in arithmetic progressions of odd difference. We also construct infinite words avo...