Van Der Waerden Numbers Research Articles

Improved Lower Bounds for Van Der Waerden Numbers

Recently, Ben Green proved that the two-color van der Waerden number ω(3, k) is bounded from below by \({k^{{b_0}\left( k \right)}}\) where \({b_0}\left( k \right) = {c_0}{\left( {{{\log \,k} \over {\log \log \,k}}} \right)^{1/3}}\). We prove a new lower bound of kb(k) with \(b\left( k \right) = {{c\log \,k} \over {\log \log \,k}}\). This is done by modifying Green’s argument, replacing a complicated result about random quadratic forms with an elementary probabilistic result.

Arithmetic progressions, quasi progressions, and Gallai-Ramsey colorings

We investigate several functions related to the r-color off-diagonal van der Waerden numbers w(m1,…,mr), where w(m1,…,mr) is the minimal integer n such that every r-coloring of {1,2,…,n} admits an mi-term arithmetic progression with all terms of color i for some i∈{1,2,…,r}. We start by giving a new lower bound for these related numbers. Next, the exact values and bounds of numbers related to quasi-progressions and mixed quasi-progression-van der Waerden numbers are given. Then, inspired by the success of graph Gallai-Ramsey theory and rainbow arithmetic progressions, we introduce the concept of Gallai-van der Waerden numbers, and obtain some exact values and bounds for these numbers, some of which are derived by the probabilistic method and the Lovász Local Lemma.

On the power of random greedy algorithms

In this paper we solve two problems of Esperet, Kang and Thomassé as well as Li concerning (i) induced bipartite subgraphs in triangle-free graphs and (ii) van der Waerden numbers. Each time random greedy algorithms allow us to go beyond the Lovász Local Lemma or alteration method used in previous work, illustrating the power of the algorithmic approach to the probabilistic method.

New lower bounds for van der Waerden numbers

AbstractWe show that there is a red-blue colouring of$[N]$with no blue 3-term arithmetic progression and no red arithmetic progression of length$e^{C(\log N)^{3/4}(\log \log N)^{1/4}}$. Consequently, the two-colour van der Waerden number$w(3,k)$is bounded below by$k^{b(k)}$, where$b(k) = c \big ( \frac {\log k}{\log \log k} \big )^{1/3}$. Previously it had been speculated, supported by data, that$w(3,k) = O(k^2)$.

A Subexponential Upper Bound for van der Waerden Numbers $W(3,k)$

We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if $\{1,\dots,N\}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$ does not contain any arithmetic progression of length $k$ then
 $$N\le \exp(O(k^{1-c}))\,.$$

Zero-sum analogues of van der Waerden’s theorem on arithmetic progressions

Let $r$ and $k$ be positive integers with $r \mid k$. Denote by $w_{\mathrm{\mathfrak{z}}}(k;r)$ the minimum integer such that every coloring $\chi:[1,w_{\mathrm{\mathfrak{z}}}(k;r)] \rightarrow \{0,1,\dots,r-1\}$ admits a $k$-term arithmetic progression $a,a+d,\dots,a+(k-1)d$ with $\sum_{j=0}^{k-1} \chi(a+jd) \equiv 0 \,(\mathrm{mod }\,r)$. We investigate these numbers as well as a mixed monochromatic/zero-sum analogue. We also present an interesting reciprocity between the van der Waerden numbers and $w_{\mathrm{\mathfrak{z}}}(k;r)$.

A novel SAT solver for the Van der Waerden numbers

This paper introduces a new efficient satisfiability problem (SAT) solver, negative-literal Van der Waerden numbers SAT solver (NegVanSAT). It is a modification of thewell-known SAT solver MINISAT where the constructor of the literals has been adjusted to start with the negated literals first. It reduces the calculations needed to solve a problem. The NegVanSAT is specifically designed for solving the satisfiability problem of finding Van der Waerden numbers, which are known to be very difficult to compute. Comparisons between the MINISAT and the proposed NegVanSAT show that the latter outperforms the MINISAT in finding many of them.

Monochromatic Hilbert Cubes and Arithmetic Progressions

The Van der Waerden number $W(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$–colored there exists a monochromatic arithmetic progression of length $k$. Similarly, the Hilbert cube number $h(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$–colored there exists a monochromatic affine $k$–cube, that is, a set of the form$$\left\{x_0 + \sum_{b \in B} b : B \subseteq A\right\}$$ for some $|A|=k$ and $x_0 \in \mathbb{Z}$.
 We show the following relation between the Hilbert cube number and the Van der Waerden number. Let $k \geqslant 3$ be an integer. Then for every $\epsilon >0$, there is a $c > 0$ such that $$h(k,4) \geqslant \min\{W(\lfloor c k^2\rfloor, 2), 2^{k^{2.5-\epsilon}}\}.$$ Thus we improve upon state of the art lower bounds for $h(k,4)$ conditional on $W(k,2)$ being significantly larger than $2^k$. In the other direction, this shows that if the Hilbert cube number is close to its state of the art lower bounds, then $W(k,2)$ is at most doubly exponential in $k$.
 We also show the optimal result that for any Sidon set $A \subset \mathbb{Z}$, one has $$\left|\left\{\sum_{b \in B} b : B \subseteq A\right\}\right| = \Omega( |A|^3) .$$

Proof of van der Waerden’s Theorem in Nine Figures

This note contains a proof of van der Waerden’s theorem, “one of the most elegant pieces of mathematics ever produced,” in nine figures. The proof follows van der Waerden’s original idea to establish the existence of what are now called van der Waerden numbers by using double induction. It also contains ideas and terminology introduced by I. Leader and T. Tao.

Some results on a class of mixed van der Waerden numbers

The mixed van der Waerden number ${w}(k_1, \ldots ,k_r;r)$ is the least positive integer $n$ such that every $r$-coloring of $[1,n]$ admits a monochromatic arithmetic progression of length $k_i$, for at least one $i$. We denote by ${w}_2(k;r)$ the case in which $k_1=\cdots =k_{r-1}=2$ and $k_r=k$. For $k \le r$, we give upper and lower bounds for ${w}_2(k;r)$, also indicating cases when these bounds are achieved. We determine exact values in the cases where $(k,r) \in \{(p,p),(p,p+1),(p+1,p+1)\}$ and give bounds in the cases where $(k,r) \in \{(p,p+2),(p+2,p+2)\}$, for primes $p$. We provide a table of values for the cases $k \le r$ with $3 \le k \le 10$ and for several values of $r$, correcting some known values.

A new lower bound for van der Waerden numbers

In this paper we prove a new recurrence relation on the van der Waerden numbers, w(r,k). In particular, if p is a prime and p≤k then w(r,k)>p⋅wr−rp,k−1. This recurrence gives the lower bound w(r,p+1)>pr−12p when r≤p, which generalizes Berlekamp’s theorem on 2-colorings, and gives the best known bound for a large interval of r. The recurrence can also be used to construct explicit valid colorings, and it improves known lower bounds on small van der Waerden numbers.

Extremal problems for colorings of simple hypergraphs and applications

The paper deals with extremal problems concerning colorings of hypergraphs. By using a random recoloring algorithm we show that any n-uniform simple hypergraph H with maximum edge degree at most Δ(H)≤c⋅nrn−1, is r-colorable, where c>0 is an absolute constant. As an application of our proof technique we establish a new lower bound for the Van der Waerden number W(n,r).

Multipass greedy coloring of simple uniform hypergraphs

Let be the minimum number of edges in an n-uniform simple hypergraph that is not two colorable. We prove that . Our result generalizes to r-coloring of b-simple uniform hypergraphs. For fixed r and b we prove that a maximum vertex degree in b-simple n-uniform hypergraph that is not r-colorable must be . By trimming arguments it implies that every such graph has edges. For any fixed our techniques yield also a lower bound for van der Waerden numbers W(n, r). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

Improved algorithms for colorings of simple hypergraphs and applications

The paper deals with extremal problems concerning colorings of hypergraphs. By using a random recoloring algorithm we show that any n-uniform simple (i.e. every two distinct edges share at most one vertex) hypergraph H with maximum edge degree at mostΔ(H)⩽c⋅nrn−1 is r-colorable, where c>0 is an absolute constant.As an application of our proof technique we establish a new lower bound for Van der Waerden number W(n,r), the minimum N such that in any r-coloring of the set {1,…,N} there exists a monochromatic arithmetic progression of length n. We show thatW(n,r)>c⋅rn−1, for some absolute constant c>0.

A Set-Coloring Generalization of van der Waerden Numbers

Let m > r ≥ 1 and k1 km be integers, and C = color i 1 ≤ i ≤ m be a color set. Let W r k1 km be the smallest positive integer n such that if any integer in 1 n is colored with an r -subset of C, then there must exist an arithmetic progression of ki terms in which any integer is colored with an r -subset of C containing color i. In this paper, such a set-coloring generalization of the van der Waerden number is proposed and studied. By studying such a set-coloring generalization, van der Waerden numbers can be understood better. On the other hand, computing set-coloring van der Waerden numbers can be regarded as a new challenge.

On the van der Waerden numbers [formula omitted

In this paper we present results and conjectures on the ordinary van der Waerden numbersw(2;3,t) and on the new palindromic van der Waerden numberspdw(2;3,t). We have computed the exact value of the previously unknown number w(2;3,19)=349, and we provide new lower bounds for 20≤t≤39, where for 20≤t≤30 we conjecture these bounds to be exact. The lower bounds for w(2;3,t) with 24≤t≤30 refute the conjecture that w(2;3,t)≤t2 as suggested in Brown et al. (2008). Based on the known values of w(2;3,t), we investigate regularities to better understand the lower bounds of w(2;3,t). Motivated by such regularities, we introduce palindromic van der Waerden numbers pdw(k;t0,…,tk−1), which are defined as the ordinary numbers w(k;t0,…,tk−1), but where only palindromic solutions are considered, reading the same from both ends. Different from the situation for ordinary van der Waerden numbers, these “numbers” need actually to be pairs of numbers. We compute pdw(2;3,t) for 3≤t≤27, and we provide bounds for t≤39, which we believe to be exact for t≤35. All computations are based on SAT solving, and we discuss the various relations between SAT solving and Ramsey theory. Especially we introduce a novel (open-source) SAT solver, the tawSolver, which performs best on the SAT instances studied here, and which is actually the original DLL-solver by Davis et al. (1962), but with an efficient implementation and a modern heuristic typical for look-ahead solvers, applying the theory developed by the second author (Kullmann, 2009).

A generalization of sets without long arithmetic progressions based on Szekeres algorithm

George Szekeres described some subsets of { 1 , … , n } without arithmetic progressions of length p for odd primes p , obtained by a greedy algorithm. Let r k ( n ) denote the size of the largest subset of { 1 , … , n } without arithmetic progressions of length k . In this paper, the history of results based on the constructions by Szekeres is briefly surveyed. New inequalities for r k ( n ) and van der Waerden numbers are derived by generalizing these constructions. In particular, for any odd prime p , we prove that r p ( p 2 ) ⩾ ( p − 1 ) 2 + t p , where lim p → ∞ t p ln p = 1 .

On semi-progression van der Waerden numbers

In this note, a dynamic programming-like method is used to detect \(k\)-term semi-progression efficiently. By using this approach, we obtain some exact values and new lower bounds on semi-progression van der Waerden numbers.

Improving the Use of Cyclic Zippers in Finding Lower Bounds for van der Waerden Numbers

For integers $k$ and $l$, each greater than $1$, suppose that $p$ is a prime with $p\equiv1\,(\textrm{mod}\, k)$ and that the $k^{\rm th}$-power classes $\textrm{mod}\, p$ induce a coloring of the integer segment $[0,\, p-1]$ that admits no monochromatic occurrence of $l$ consecutive members of an arithmetic progression. Such a coloring can lead to a coloring of $[0,\,(l-1)p]$ that is similarly free of monochromatic $l$-progressions, and, hence, can give directly a lower bound for the van der Waerden number $W(k,l)$. P. R. Herwig, M. J. H. Heule, P. M. van Lambalgen, and H. van Maaren have devised a technique for splitting and "zipping" such a coloring of $[0,\, p-1]$ to yield a coloring of $[0,\,2p-1]$ which, for even values of $k$, is sometimes extendable to a coloring of $[0,\,2(l-1)p]$ where both new colorings still admit no monochromatic $l$-progressions. Here we derive a fast procedure for checking whether such a zipped coloring remains free of monochromatic $l$-progressions, effectively reducing a quadratic-time check to a linear-time check. Using this procedure we find some new lower bounds for van der Waerden numbers.

On sets without k-term arithmetic progression

For positive integers n and k, let r k ( n ) be the size of the largest subset of { 1 , 2 , … , n } without arithmetic progressions of length k. The van der Waerden number W ( k 1 , k 2 , … , k r ) is the smallest integer w such that every r-coloring of { 1 , 2 , … , w } contains a monochromatic k i -term arithmetic progression with color i for some i. In this note, an algorithm is proposed to search exact values of r k ( n ) for some k and n, and some new exact values of r k ( n ) for k = 4 , 5 , 6 , 7 , 8 are obtained. The results extend the previous ones significantly. It is also shown that r k + 1 ( 2 k 2 + 1 ) ⩾ 2 k 2 − 3 k + 3 for prime k ⩾ 3 , and three lower bounds for van der Waerden numbers are given: W ( 3 , 4 , 5 ) ⩾ 124 , W ( 5 , 8 ) ⩾ 248 , W ( 5 , 9 ) ⩾ 320 .