Independent Finite Sums forKm-Free Graphs

Recently, in conversation with Erdős, Hajnal asked whether or not for any triangle-free graphGon the vertex set N, there always exists a sequence ⦠xn⦔∞n=1so that wheneverFandHare distinct finite nonempty subsets of N, {∑n∈F xn, ∑n∈H xn} is not an edge ofG(that is,FS(⦠xn⦔∞n=1) is an independent set). We answer this question in the negative. We also show that if one replaces the assumption thatGis triangle-free by the assertion that for somem,Gcontains no complete bipartite graphKm, m, then the conclusion does hold. If for somem⩾3,Gcontains noKm, we show there exists a sequence ⦠xn⦔∞n=1so that wheneverFandHare disjoint finite nonempty subsets of N, {∑n∈F xn, ∑n∈H xn} is not an edge ofG. Both of the affirmative results are in fact valid for a graphGon an arbitrary cancellative semigroup (S, +).