Let σ(k) denote the maximum of the number of squares in a+b, . . . , a+kb as we vary over positive integers a and b. Erdős conjectured that σ(k) = o(k) which Szemerédi [30] elegantly proved as follows: If there are more than δk squares amongst the integers a+b, . . . , a+kb (where k is sufficiently large) then there exists four indices 1 ≤ i1 < i2 < i3 < i4 ≤ k in arithmetic progression such th...

We consider arithmetic progressions on Pellian equations x2 − d y2 = m, i.e. integral solutions such that the y-coordinates are in arithmetic progression. We construct explicit infinite families of d,m for which there exists a five-term arithmetic progression in the y-coordinate, and we prove the existence of infinitely many pairs d,m parametrized by the points of an elliptic curve of positive ...

Given an integer r ≥ 2 and a number υ ∈ (0, 1], consider the collection of all subsets of Z/rZ having at least υr elements. Among the sets in this collection, suppose S is any one having the minimal number of three-term arithmetic progressions, where in our terminology a three-term arithmetic progression is a triple (x, y, z) ∈ S3 satisfying x + y ≡ 2z (mod r). Note that this includes trivial p...

Given an integer q ≥ 2 and a number θ ∈ (0, 1], consider the collection of all subsets of Zq := Z/qZ having at least θq elements. Among the sets in this collection, suppose S is any one having the minimal number of three-term arithmetic progressions, where in our terminology a three-term arithmetic progression is a triple (x, y, z) ∈ S3 satisfying x + y ≡ 2z (mod q). Note that this includes tri...

A theorem of Dekking in the combinatorics of words implies that there exists an injective order-preserving transformation f : {0, 1, . . . , n} → {0, 1, . . . , 2n} with the restriction f(i + 1) ≤ f(i) + 2 such that for every 5-term arithmetic progression P its image f(P ) is not an arithmetic progression. In this paper we consider symmetric sets in place of arithmetic progressions and prove lo...

A connected graph G is said to be (a, d)-antimagic, for some positive integers a and d, if its edges admit a labeling by the integers 1,2, ... , IE( G) I such that the induced vertex labels consist of an arithmetic progression with the first term a and the common difference d. In this paper we prove that the generalized Petersen graph P(n,2) is (3n2+6, 3)-antimagic for n == 0 (mod 4), n ~ B.

The anti-van der Waerden number, denoted by aw([n], k), is the smallest r such that every exact r-coloring of [n] contains a rainbow k-term arithmetic progression. Butler et al. showed that dlog3 ne + 2 6 aw([n], 3) 6 dlog2 ne + 1, and conjectured that there exists a constant C such that aw([n], 3) 6 dlog3 ne + C. In this paper, we show this conjecture is true by determining aw([n], 3) for all ...

The purpose of this paper is to provide a simple and self-contained exposition of the celebrated Roth's theorem on arithmetic progressions of length three. The original result is proved in [Roth53], while the proof given below is very similar to the exposition of Roth's original argument given in [GRS1990]. Definition. We say that a subset of positive integers A has positive upper density if (1...