Infinite graphs that do not contain cycles of length four

We construct a countable infinite graph G that does not contain cycles of length four having the property that the sequence of graphs $G_n$ induced by the first $n$ vertices has minimum degree $\delta(G_n)> n^{\sqrt{2}-1+o(1)}$.Comment: This paper has been withdrawn because we have found an easier proof of the resul

Combinatorial problems in finite fields and Sidon sets

We use Sidon sets to present an elementary method to study some combinatorial problems in finite fields, such as sum product estimates, solubility of some equations and distribution of sequences in small intervals. We obtain classic and more recent results avoiding the use of exponential sums, the usual tool to deal with these problems.Comment: 13 page

Dense sets of integers with prescribed representation functions

Let A be a set of integers and let h \geq 2. For every integer n, let r_{A, h}(n) denote the number of representations of n in the form n=a_1+...+a_h, where a_1,...,a_h belong to the set A, and a_1\leq ... \leq a_h. The function r_{A,h} from the integers Z to the nonnegative integers N_0 U {\infty} is called the representation function of order h for the set A. We prove that every function f from Z to N_0 U {\infty} satisfying liminf_{|n|->\infty} f (n)\geq g is the representation function of order h for some sequence A of integers, and that A can be constructed so that it increases "almost" as slowly as any given B_h[g] sequence. In particular, for every epsilon >0 and g \geq g(h,epsilon), we can construct a sequence A satisfying r_{A,h}=f and A(x)\gg x^{(1/h)-epsilon}.Comment: 10 page

Lattice points on circles, squares in arithmetic progressions and sumsets of squares

Rudin conjectured that there are never more than c N^(1/2) squares in an arithmetic progression of length N. Motivated by this surprisingly difficult problem we formulate more than twenty conjectures in harmonic analysis, analytic number theory, arithmetic geometry, discrete geometry and additive combinatorics (some old and some new) which each, if true, would shed light on Rudin's conjecture.Comment: 21 pages, preliminary version. Comments welcom