Some Results in Extremal Combinatorics

Extremal Combinatorics is one of the central and heavily contributed areas in discrete mathematics, and has seen an outstanding growth during the last few decades. In general, it deals with problems regarding determination and/or estimation of the maximum or the minimum size of a combinatorial structure that satisfies a certain combinatorial property. Problems in Extremal Combinatorics are often related to theoretical computer science, number theory, geometry, and information theory. In this thesis, we work on some well-known problems (and on their variants) in Extremal Combinatorics concerning the set of integers as the combinatorial structure. The van der Waerden number w(k;t_0,t_1,...,t_{k-1}) is the smallest positive integer n such that every k-colouring of 1, 2, . . . , n contains a monochromatic arithmetic progression of length t_j for some colour j in {0,1,...,k-1}. We have determined five new exact values with k=2 and conjectured several van der Waerden numbers of the form w(2;s,t), based on which we have formulated a polynomial upper-bound-conjecture of w(2; s, t) with fixed s. We have provided an efficient SAT encoding for van der Waerden numbers with k>=3 and computed three new van der Waerden numbers using that encoding. We have also devised an efficient problem-specific backtracking algorithm and computed twenty-five new van der Waerden numbers with k>=3 using that algorithm. We have proven some counting properties of arithmetic progressions and some unimodality properties of sequences regarding arithmetic progressions. We have generalized Szekeres’ conjecture on the size of the largest sub-sequence of 1, 2, . . . , n without an arithmetic progression of length k for specific k and n; and provided a construction for the lower bound corresponding to the generalized conjecture. A Strict Schur number S(h,k) is the smallest positive integer n such that every 2-colouring of 1,2,...,n has either a blue solution to x_1 +x_2 +···+x_{h-1} = x_h where x_1 < x_2 < ··· < x_h, or a red solution to x_1+x_2+···+x_{k-1} =x_k where x_1 <x_2 <···<x_k. We have proven the exact formula for S(3, k)

New Lower Bounds for van der Waerden Numbers Using Distributed Computing

This paper provides new lower bounds for van der Waerden numbers. The number $W(k,r)$ is defined to be the smallest integer $n$ for which any $r$-coloring of the integers $0 \ldots, n-1$ admits monochromatic arithmetic progression of length $k$; its existence is implied by van der Waerden's Theorem. We exhibit $r$-colorings of $0\ldots n-1$ that do not contain monochromatic arithmetic progressions of length $k$ to prove that $W(k, r)>n$. These colorings are constructed using existing techniques. Rabung's method, given a prime $p$ and a primitive root $\rho$, applies a color given by the discrete logarithm base $\rho$ mod $r$ and concatenates $k-1$ copies. We also used Herwig et al's Cyclic Zipper Method, which doubles or quadruples the length of a coloring, with the faster check of Rabung and Lotts. We were able to check larger primes than previous results, employing around 2 teraflops of computing power for 12 months through distributed computing by over 500 volunteers. This allowed us to check all primes through 950 million, compared to 10 million by Rabung and Lotts. Our lower bounds appear to grow roughly exponentially in $k$. Given that these constructions produce tight lower bounds for known van der Waerden numbers, this data suggests that exact van der Waerden Numbers grow exponentially in $k$ with ratio $r$ asymptotically, which is a new conjecture, according to Graham.Comment: 8 pages, 1 figure. This version reflects new results and reader comment

On the van der Waerden numbers w(2;3,t)

We present results and conjectures on the van der Waerden numbers w(2;3,t) and on the new palindromic van der Waerden numbers pdw(2;3,t). We have computed the new number w(2;3,19) = 349, and we provide lower bounds for 20 <= t <= 39, where for t <= 30 we conjecture these lower bounds to be exact. The lower bounds for 24 <= t <= 30 refute the conjecture that w(2;3,t) <= t^2, and we present an improved conjecture. We also investigate regularities in the good partitions (certificates) to better understand the lower bounds. Motivated by such reglarities, we introduce *palindromic van der Waerden numbers* pdw(k; t_0,...,t_{k-1}), defined as ordinary van der Waerden numbers w(k; t_0,...,t_{k-1}), however only allowing palindromic solutions (good partitions), defined as 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 lower bounds, which we conjecture 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, but with an efficient implementation and a modern heuristic typical for look-ahead solvers (applying the theory developed in the SAT handbook article of the second author).Comment: Second version 25 pages, updates of numerical data, improved formulations, and extended discussions on SAT. Third version 42 pages, with SAT solver data (especially for new SAT solver) and improved representation. Fourth version 47 pages, with updates and added explanation

Lower Bounds on the van der Waerden Numbers: Randomized- and Deterministic-Constructive

The van der Waerden number W(k,2) is the smallest integer n such that every 2-coloring of 1 to n has a monochromatic arithmetic progression of length k. The existence of such an n for any k is due to van der Waerden but known upper bounds on W(k,2) are enormous. Much effort was put into developing lower bounds on W(k,2). Most of these lower bound proofs employ the probabilistic method often in combination with the Lov\'asz Local Lemma. While these proofs show the existence of a 2-coloring that has no monochromatic arithmetic progression of length k they provide no efficient algorithm to find such a coloring. These kind of proofs are often informally called nonconstructive in contrast to constructive proofs that provide an efficient algorithm. This paper clarifies these notions and gives definitions for deterministic- and randomized-constructive proofs as different types of constructive proofs. We then survey the literature on lower bounds on W(k,2) in this light. We show how known nonconstructive lower bound proofs based on the Lov\'asz Local Lemma can be made randomized-constructive using the recent algorithms of Moser and Tardos. We also use a derandomization of Chandrasekaran, Goyal and Haeupler to transform these proofs into deterministic-constructive proofs. We provide greatly simplified and fully self-contained proofs and descriptions for these algorithms

Using the Incompressibility Method to obtain Local Lemma results for Ramsey-type Problems

We reveal a connection between the incompressibility method and the Lovasz local lemma in the context of Ramsey theory. We obtain bounds by repeatedly encoding objects of interest and thereby compressing strings. The method is demonstrated on the example of van der Waerden numbers. It applies to lower bounds of Ramsey numbers, large transitive subtournaments and other Ramsey phenomena as well.Comment: 8 pages, 1 figur