Let h≥2 and let A=(A1,…,Ah) be an h-tuple of sets integers. For nonzero integers c1,…,ch, consider the linear form φ=c1x1+c2x2+⋯+chxh. The representation function RA,φ(n) counts number h-tuples (a1,…,ah)∈A1×⋯×Ah such that φ(a1,…,ah)=n. A is a φ-Sidon system multiplicity g if RA,φ(n)≤g for all n∈Z. every positive integer g, Fφ,g(n) denote largest q there exists withAi⊆[1,n]and|Ai|=q i=1,…,h. It ...

We prove that every 4-coloring of [n] with each color class having cardinality more than n+1 6 contains a rainbow solution to x + y = z + w, and that this bound is tight.

The correspondence between perfect difference sets and transitive projective planes is well-known. We observe that all known dense (i.e., close to square-root size) Sidon subsets of abelian groups come from through a similar construction. classify the arising in this manner desarguesian find essentially no new examples, but there are many further examples nondesarguesian planes. conjecture aris...

A set of nonnegative integers is called a Sidon if there no 4-tuple, i.e., (a, b, c, d) in with + b = c d and {a, b} ∩ {c, d} ∅. Cameron Erdős proposed the problem determining number sets [n]. Results Kohayakawa, Lee, Rodl Samotij, Saxton Thomason have established that between 2 (1.16+o(1))√n (6.442+o(1))√n . An α-generalized [n] at most α 4-tuples. One way to extend estimate We show (n/ log4 n...

A Bh set (or Sidon set of order h) in an Abelian group G is any subset {b0, b1, . . . , bn} ⊂ G with the property that all the sums bi1 + · · ·+ bih are different up to the order of the summands. Let φ(h, n) denote the order of the smallest Abelian group containing a Bh set of cardinality n+1. It is shown that, as h → ∞ and n is kept fixed, φ(h, n) ∼ 1 n! δl(△) h n , where δl(△ ) is the lattice...