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

On variables with few occurrences in conjunctive normal forms

We consider the question of the existence of variables with few occurrences in boolean conjunctive normal forms (clause-sets). Let mvd(F) for a clause-set F denote the minimal variable-degree, the minimum of the number of occurrences of variables. Our main result is an upper bound mvd(F) <= nM(surp(F)) <= surp(F) + 1 + log_2(surp(F)) for lean clause-sets F in dependency on the surplus surp(F). - Lean clause-sets, defined as having no non-trivial autarkies, generalise minimally unsatisfiable clause-sets. - For the surplus we have surp(F) <= delta(F) = c(F) - n(F), using the deficiency delta(F) of clause-sets, the difference between the number of clauses and the number of variables. - nM(k) is the k-th "non-Mersenne" number, skipping in the sequence of natural numbers all numbers of the form 2^n - 1. We conjecture that this bound is nearly precise for minimally unsatisfiable clause-sets. As an application of the upper bound we obtain that (arbitrary!) clause-sets F with mvd(F) > nM(surp(F)) must have a non-trivial autarky (so clauses can be removed satisfiability-equivalently by an assignment satisfying some clauses and not touching the other clauses). It is open whether such an autarky can be found in polynomial time. As a future application we discuss the classification of minimally unsatisfiable clause-sets depending on the deficiency.Comment: 14 pages. Revision contains more explanations, and more information regarding the sharpness of the boun

Finding Periodic Apartments via Boolean Satisfiability and Orderly Generation

Peer reviewe

Constraint satisfaction problems in clausal form

This is the report-version of a mini-series of two articles on the foundations of satisfiability of conjunctive normal forms with non-boolean variables, to appear in Fundamenta Informaticae, 2011. These two parts are here bundled in one report, each part yielding a chapter. Generalised conjunctive normal forms are considered, allowing literals of the form "variable not-equal value". The first part sets the foundations for the theory of autarkies, with emphasise on matching autarkies. Main results concern various polynomial time results in dependency on the deficiency. The second part considers translations to boolean clause-sets and irredundancy as well as minimal unsatisfiability. Main results concern classification of minimally unsatisfiable clause-sets and the relations to the hermitian rank of graphs. Both parts contain also discussions of many open problems.Comment: 91 pages, to appear in Fundamenta Informaticae, 2011, as Constraint satisfaction problems in clausal form I: Autarkies and deficiency, Constraint satisfaction problems in clausal form II: Minimal unsatisfiability and conflict structur

Finding Periodic Apartments : A Computational Study of Hyperbolic Buildings

This thesis presents a computational study of a fundamental open conjecture in geometric group theory using an intricate combination of Boolean Satisfiability and orderly generation. In particular, we focus on Gromov’s subgroup conjecture (GSC), which states that “each one-ended hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed surface of genus at least 2”. Several classes of groups have been shown to satisfy GSC, but the status of non-right-angled groups with regard to GSC is presently unknown, and may provide counterexamples to the conjecture. With this in mind Kangaslampi and Vdovina constructed 23 such groups utilizing the theory of hyperbolic buildings [International Journal of Algebra and Computation, vol. 20, no. 4, pp. 591–603, 2010], and ran an exhaustive computational analysis of surface subgroups of genus 2 arising from so-called periodic apartments [Experimental Mathematics, vol. 26, no. 1, pp. 54–61, 2017]. While they were able to rule out 5 of the 23 groups as potential counterexamples to GSC, they reported that their computational approach does not scale to genera higher than 2. We extend the work of Kangaslampi and Vdovina by developing two new approaches to analyzing the subgroups arising from periodic apartments in the 23 groups utilizing different combinations of SAT solving and orderly generation. We develop novel SAT encodings and a specialized orderly algorithm for the approaches, and perform an exhaustive analysis (over the 23 groups) of the genus 3 subgroups arising from periodic apartments. With the aid of massively parallel computation we also exhaust the case of genus 4. As a result we rule out 4 additional groups as counterexamples to GSC leaving 14 of the 23 groups for further inspection. In addition to this our approach provides an independent verification of the genus 2 results reported by Kangaslampi and Vdovina

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)

Constraint Satisfaction Techniques for Combinatorial Problems

The last two decades have seen extraordinary advances in tools and techniques for constraint satisfaction. These advances have in turn created great interest in their industrial applications. As a result, tools and techniques are often tailored to meet the needs of industrial applications out of the box. We claim that in the case of abstract combinatorial problems in discrete mathematics, the standard tools and techniques require special considerations in order to be applied effectively. The main objective of this thesis is to help researchers in discrete mathematics weave through the landscape of constraint satisfaction techniques in order to pick the right tool for the job. We consider constraint satisfaction paradigms like satisfiability of Boolean formulas and answer set programming, and techniques like symmetry breaking. Our contributions range from theoretical results to practical issues regarding tool applications to combinatorial problems. We prove search-versus-decision complexity results for problems about backbones and backdoors of Boolean formulas. We consider applications of constraint satisfaction techniques to problems in graph arrowing (specifically in Ramsey and Folkman theory) and computational social choice. Our contributions show how applying constraint satisfaction techniques to abstract combinatorial problems poses additional challenges. We show how these challenges can be addressed. Additionally, we consider the issue of trusting the results of applying constraint satisfaction techniques to combinatorial problems by relying on verified computations