On (k, l)-sets in cyclic groups of odd prime order

Abstract

Let A be a finite Abelian group written additively. For two positive integers k, l with k ≠ l, we say that a subset S ⊂ A is of type (k, l) or is a (k, l) -set if the equation x1 + x2 + … + xk − xk+1−… − xk+1 = 0 has no solution in the set S. In this paper we determine the largest possible cardinality of a (k, l)-set of the cyclic group ℤP where p is an odd prime. We also determine the number of (k, l)-sets of ℤp which are in arithmetic progression and have maximum cardinality.

ON THE MAXIMUM SIZE OF A (k,l)-SUM-FREE SUBSET OF AN ABELIAN GROUP

“…A pleasing answer is given by Bier and Chin [5] for the case when k ≥ 3 and G ∼ = Z p where p is an odd prime: in this case A is an arithmetic progression. The same answer was given by Diananda and Yap [13] earlier for the case when (k, l) = (2, 1) (that is, when A is sum-free) and G ∼ = Z p with p not congruent to 1 mod 3; however, for p = 3m + 1 the set A = {m, m + 2, m + 3, .…”

A new critical pair theorem applied to sum-free sets in Abelian groups

“…Only the case of (k, l)-free subsets of Z/pZ (p prime) is answered. In that case, both the value [2] of λ k,l (Z/pZ) and the structure [21] of the (k, l)-free subsets of Z/pZ with that cardinality are known.…”

“…Theorem D [2,21]. Let p be an odd prime and let k, l be positive integers not congruent modulo p and which satisfy max(k, l) ≥ 3.…”

The Maximum Size of -Sum-Free Sets in Cyclic groups

“…The first result for general k and l was given by Bier and Chin. Theorem 1.3 (Bier and Chin, 2001; see [4]). Let p be a positive prime.…”