3-term Arithmetic Progressions Research Articles

On the maximal number of 3-term arithmetic progressions in subsets of ℤ/p ℤ

Let α ∈ [0, 1] be a real number. Ernie Croot (Canad. Math. Bull. 51 (2008) 47–56) showed that the quantity maxA # (3-term arithmetic progressions in A)/p2, where A ranges over all subsets of ℤ/pℤ of size at most α p, tends to a limit as p → ∞ through primes. Writing c(α) for this limit, we show that c(α)=α2/2 provided that α is smaller than some absolute constant. In fact, we prove rather more, establishing a structure theorem for sets having the maximal number of 3-term progressions amongst all subsets of ℤ/p ℤ of cardinality m, provided that m < cp.

Bounds on some van der Waerden numbers

For positive integers s and k 1 , k 2 , … , k s , the van der Waerden number w ( k 1 , k 2 , … , k s ; s ) is the minimum integer n such that for every s-coloring of set { 1 , 2 , … , n } , with colors 1 , 2 , … , s , there is a k i -term arithmetic progression of color i for some i. We give an asymptotic lower bound for w ( k , m ; 2 ) for fixed m. We include a table of values of w ( k , 3 ; 2 ) that are very close to this lower bound for m = 3 . We also give a lower bound for w ( k , k , … , k ; s ) that slightly improves previously-known bounds. Upper bounds for w ( k , 4 ; 2 ) and w ( 4 , 4 , … , 4 ; s ) are also provided.

On the asymptotic minimum number of monochromatic 3-term arithmetic progressions

Let V ( n ) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-coloring of { 1 , 2 , … , n } . We show that 1675 32 768 n 2 ( 1 + o ( 1 ) ) ⩽ V ( n ) ⩽ 117 2192 n 2 ( 1 + o ( 1 ) ) . As a consequence, we find that V ( n ) is strictly greater than the corresponding number for Schur triples (which is 1 22 n 2 ( 1 + o ( 1 ) ) ). Additionally, we disprove the conjecture that V ( n ) = 1 16 n 2 ( 1 + o ( 1 ) ) as well as a more general conjecture.

Some Ramsey and anti–Ramsey results in finite groups

Abstract We give a combinatorial argument which allows one to prove some Ramsey and anti Ramsey results in colorings of finite groups. We discuss some applications to the study of monochromatic and rainbow structures in groups, including solutions to the Schur equation x y = z and 3–term arithmetic progressions.

Coloured solutions of equations in finite groups

In this article, we consider the relations between colourings and some equations in finite groups. We will express relations linking the numbers of the differently coloured solutions of an equation that depend only on the cardinality of the colouring and not on the distribution of the colours. This gives a link between Ramsey theory that investigates the existence of monochromatic solutions and what is now called anti-Ramsey theory that investigates the existence of rainbow solutions. Both theories are in expansion. We will apply these results to the counting of rainbow 3-term arithmetic progressions in any abelian group with equinumerous three-colouring and to the counting of points on a conic defined on a finite field. We will end by discussing the generalised case of a system of equations.

Arithmetic Progressions in Sets with Small Sumsets

We present an elementary proof that if $A$ is a finite set of numbers, and the sumset $A+_GA$ is small, $|A+_GA|\leq c|A|$, along a dense graph $G$, then $A$ contains $k$-term arithmetic progressions.

Roth’s theorem in the primes

We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood majorant property. We derive this by giving a new proof of a rather more general result of Bourgain which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a restriction theorem for the primes.

Almost disjoint families of 3-term arithmetic progressions

We obtain upper and lower bounds for the size of a largest family of 3-term arithmetic progressions contained in [ 0 , n - 1 ] , no two of which intersect in more than one point. Such a family consists of just under a half of all of the 3-term arithmetic progressions contained in [ 0 , n - 1 ] .

On some Rado numbers for generalized arithmetic progressions

The 2-color Rado number for the equation x 1+ x 2−2 x 3= c, which for each constant c∈ Z we denote by S 1( c), is the least integer, if it exists, such that every 2-coloring, Δ : [1,S 1(c)]→{0,1} , of the natural numbers admits a monochromatic solution to x 1+ x 2−2 x 3= c, and otherwise S 1( c)=∞. We determine the 2-color Rado number for the equation x 1+ x 2−2 x 3= c, when additional inequality restraints on the variables are added. In particular, the case where we require x 2< x 3< x 1, is a generalization of the 3-term arithmetic progression; and the work done here improves previously established upper bounds to an exact value.

Rainbow Arithmetic Progressions and Anti-Ramsey Results

The van der Waerden theorem in Ramsey theory states that, for every k and t and sufficiently large N, every k-colouring of [N] contains a monochromatic arithmetic progression of length t. Motivated by this result, Radoičić conjectured that every equinumerous 3-colouring of [3n] contains a 3-term rainbow arithmetic progression, i.e., an arithmetic progression whose terms are coloured with distinct colours. In this paper, we prove that every 3-colouring of the set of natural numbers for which each colour class has density more than 1/6, contains a 3-term rainbow arithmetic progression. We also prove similar results for colourings of . Finally, we give a general perspective on other anti-Ramsey-type problems that can be considered.

Triangle Free Sets and Arithmetic Progressions &ndash; Two Pisier Type Problems

Let ${\cal P}_f({\bf N})$ be the set of finite nonempty subsets of ${\bf N}$ and for $F,G\in{\cal P}_f({\bf N})$ write $F &lt; G$ when $\max F &lt; \min G$. Let $X=\{(F,G):F,G\in{\cal P}_f({\bf N})$ and $F &lt; G\}$. A triangle in $X$ is a set of the form $\{(F\cup H,G),(F,G),(F,H\cup G)\}$ where $F &lt; H &lt; G$. Motivated by a question of Erdős, Nešetríl, and Rödl regarding three term arithmetic progressions, we show that any finite subset $Y$ of $X$ contains a relatively large triangle free subset. Exact values are obtained for the largest triangle free sets which can be guaranteed to exist in any set $Y\subseteq X$ with $n$ elements for all $n\leq 14$.

Monochromatic Homothetic Copies of {1, 1 + s, 1 + s + t}

AbstractFor positive integers s and t, let f(s, t) denote the smallest positive integer N such that every 2-colouring of [1, N] = {1, 2,...,N} has a monochromatic homothetic copy of {1, 1 + s, 1 + s + t}.We show that f (s, t) = 4(s + t) + 1 whenever s/g and t/g are not congruent to 0 (modulo 4), where g = gcd(s, t). This can be viewed as a generalization of part of van der Waerden’s theorem on arithmetic progressions, since the 3-term arithmetic progressions are the homothetic copies of {1, 1 + 1, 1 + 1 + t}. We also show that f (s, t) = 4(s + t) + 1 in many other cases (for example, whenever s &gt; 2t &gt; 2 and t does not divide s), and that f (s, t) ≤ 4 (s + t) + 1 for all s, t.Thus the set of homothetic copies of {1, 1 + s, 1 + s + t} is a set of triples with a particularly simple Ramsey function (at least for the case of two colours), and one wonders what other “natural” sets of triples, quadruples, etc., have simple (or easily estimated) Ramsey functions.

On subsets of finite abelian groups with no 3-term arithmetic progressions

Let G be a finite abelian group of odd order and let D(G) denote the maximal cardinality of a subset A ⊂ G which does not contain a 3-term arithmetic progression. It is shown that D( Z k 1 ⊕ ⋯ ⊕ Z k n ) ⩽ 2(( k 1 ⋯ k n / n). Together with results of Szemerédi and Heath-Brown it implies that there exists a β > 0 such that D( G) = O(∥ G∥/(log ∥ G∥) β ) for all G.

Pythagorean ratios in arithmetic progression, part II. Four Pythagorean ratios

As in [3] let {a, b}designate the Pythagorean ratio (a2 − b2)/2ab between the sides of a rational right angled triangle. The principal result of [3] is that {a, b}is the arithmetic mean of two Pythagorean ratios, and hence is the middle term of a three term arithmetic progression, if and only if a /b is the geometric mean of two Pythagorean ratios. Here in Part II we study sets of four Pythagorean ratios in arithmetic progression. We show that sets of four in consecutive places in an arithmetic progression are closely related to sets of four in the first, second, third and fifth places in a progression; any one of the former sets determines two of the latter sets, and either one of the latter sets determines the other and the former. We construct an infinite sequence of sets of four ratios in consecutive places of arithmetic progressions, the last term of each set being the first term of the next set. These sets are related to solutions of the Diophantine equations r2 = 5p2q2 ± 4(p4 − 2q4). Computer searches, in addition to exhibiting enough members of this sequence to enable us to identify it, also exhibited two sets which do not belong to this sequence.

Linear equations in primes

§1. Introduction. The literature on solving a system of linear equations in primes is quite limited, although the multi-dimensional Hardy-Littlewood method certainly provides an approach to this problem. The Goldbach- Vinogradov theorem and van der Corput's proof of the existence of infinitely many three term arithmetic progressions in primes are two particular results in the special case of only one equation. Recently Liu and Tsang [4] studied this case in full generality and obtained a result with excellent uniformity in the coefficients. Almost no other general result has appeared so far, due probably to the fact that such a theorem is clumsy to state.

Small Sets which meet all the n -Term Arithmetic Progressions in the Interval [1, n2

Small Sets which meet all the <i>n</i> -Term Arithmetic Progressions in the Interval [1, n²

Small sets which meet all the k(n)-term arithmetic progressions in the interval [1, n

For given n, k, the minimum cardinal of any subset B of [1, n] which meets all of the k-term arithmetic progressions contained in [1, n] is denoted by ƒ(n, k). We show, answering questions raised by Professor P. Erdös, that ƒ(n, n ϵ) < C · n 1−ϵ for some constant C (where C depends on ϵ), and that ƒ(n, log n) = o(n) . We also discuss the behavior of ƒ(p 2, p) when p is a prime, and we give a simple lower bound for the function associated with Szemerédi's theorem.

On subsets of abelian groups with no 3-term arithmetic progression

A short proof of the following result of Brown and Buhler is given: For any ϵ > 0 there exists n 0 = n 0( ϵ) such that if A is an abelian group of odd order | A| > n 0 and B ⊆ A with | B| > ϵ| A|, then B must contain three distinct elements x, y, z satisfying x + y = 2 z.

A canonical restricted version of van der waerden’s theorem

It is shown that there is a subsetS of integers containing no (k+1)-term arithmetic progression such that if the elements ofS are arbitrarily colored (any number of colors),S will contain ak-term arithmetic progression for which all of its terms have the same color, or all have distinct colors.

Variations on a game

This survey paper contains several nice results of the authors on special positional games called amoeba Two players, Maker and Breaker, move alternately with Maker going first. In each turn they occupy previously unoccupied vertices of a hypergraph. In the game Maker's aim is to occupy all vertices of some edge of the hypergraph before Breaker can do it. In the game Braker's aim is to prevent Maker from achieving her goal without looking at his own pieces. For example, in the Van der Waerden game the winning sets are the $n$ term arithmetic progressions for some fixed $n>1$. We discuss this, and several other games, when can Maker and when can Breaker win. We show that the weak games have compactness property, and we give counterexamples for strong games. We also consider infinite games where the players keep on taking moves until there are any unoccupied vertex in an infinite grap.