Abstract We consider two recent conjectures made by Harrington, Henninger-Voss, Karhadkar, Robinson and Wong concerning relationships between the sum index, difference index exclusive number of graphs. One conjecture posits an exact relationship first invariants; we show that in fact predicted value may be arbitrarily far from truth either direction. In process establish some new bounds on both...

lie in finitely many proper subspaces of Q. As an application of the subspace theorem Schmidt [16] described all norm form equations that have finitely many solutions. The subspace theorem has been further developed by Schlickewei [11, 12] and is proved in it’s most general form by Evertse and Schlickewei [3] (see also [13]). These investigations led to many applications, e.g. to the finiteness...

A set S of positive integers has distinct subset sums if there are 2|S| distinct elements of the set {∑ x∈X x : X ⊂ S } . Let f(n) = min{maxS : |S| = n and S has distinct subset sums}. Erdős conjectured f(n) ≥ c2n for some constant c. We give a construction that yields f(n) < 0.22002 · 2n for n sufficiently large. This now stands as the best known upper bound on f(n).

A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and the sums labels taken over distinct hyperedges are distinct. Let s(H) be smallest integer N such that there sum-distinguishing H with each label at most N. The largest value all hypergraphs on n vertices m denoted s(n,m). We prove s(n,m) almost-quadratic in as long not too large. More precisely, following h...

For given positive integers m and n, we consider the frequency of representations of m n as a sum of unit fractions.