We show that √ k2k/2 is, roughly, the threshold where, under mild conditions, on one side almost every coloring contains a monochromatic k-term arithmetic progression, while on the other side, there are almost no such colorings.

Suppose a, b, c, and d are rational numbers such that a2, b2, c2, and d2 form an arithmetic progression: the differences b2−a2, c2−b2, and d2−c2 are equal. One possibility is that the arithmetic progression is constant: a2, a2, a2, a2. Are there arithmetic progressions of four rational squares which are not constant? This question was first raised by Fermat in 1640. There are no such progressio...

In this note we are interested in the problem of whether or not every increasing sequence of positive integers x1x2x3 · · · with bounded gaps must contain a double 3-term arithmetic progression, i.e., three terms xi, xj , and xk such that i+ k = 2j and xi+xk = 2xj . We consider a few variations of the problem, discuss some related properties of double arithmetic progressions, and present severa...

Several renowned open conjectures in combinatorics and number theory involve arithmetic progressions. Van der Waerden famously proved in 1927 that for each positive integer k there exists a least positive integer w(k) such that any 2-coloring of 1, . . . , w(k) produces a monochromatic k-term arithmetic progression. The best known upper bound for w(k) is due to Gowers and is quite large. Ron Gr...

In this note we are interested in the problem of whether or not every increasing sequence of positive integers x1x2x3 · · · with bounded gaps must contain a double 3-term arithmetic progression, i.e., three terms xi, x j, and xk such that i+k = 2 j and xi+xk = 2x j. We consider a few variations of the problem, discuss some related properties of double arithmetic progressions, and present severa...

We give a short proof for the following result on the distribution of three-term arithmetic progressions in sparse subsets of Fp : for every α > 0 there exists a constant C = C(α) such that the following holds for all r ≥ Cp and for almost all sets R of size r of Fp . Let A be any subset of R of size at least αr, then A contains a non-trivial three-term arithmetic progression. This is an analog...

Let p(n) denote the number of partitions of a non-negative integer n. A well-known conjecture asserts that every arithmetic progression contains infinitely many integers M for which p(M) is odd, as well as infinitely many integers N for which p(N) is even (see Subbarao [22]). From the works of various authors, this conjecture has been verified for every arithmetic progression with modulus t whe...

Let α, σ > 0 and let A and S be subsets of a finite abelian group G of densities α and σ independent of |G|, respectively. Without additional restrictions A need not contain a 3-term arithmetic progression whose common gap is in S. What is then the least integer k ≥ 2 for which there exists an η = η(α, σ) such that ‖S‖Uk(G) ≤ η implies that A contains a non-trivial 3-term arithmetic progression...