Independent Finite Sums forKm-Free Graphs

Recently, in conversation with Erdős, Hajnal asked whether or not for any triangle-free graph G on the vertex set N, there always exists a sequence 〈xn〉n=1 so that whenever F and H are distinct finite nonempty subsets of N, {Σn∈F xn,Σn∈H xn} is not an edge of G (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 that G is triangle-free by the assertion that for some m, G contains no complete bipartite graph Km,m, then the conclusion does hold. If for some m ≥ 3, G contains no Km, we show there exists a sequence 〈xn〉n=1 so that whenever F and H are disjoint finite nonempty subsets of N, {Σn∈F xn,Σn∈H xn} is not an edge of G. Both of the affirmative results are in fact valid for a graph G on an arbitrary cancellative semigroup (S,+).