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

Combinatorics of finite sets

Let $\lbrack n\rbrack = \{1,2,\..., n\},A$ and let 2\sp{\lbrack n\rbrack} represent the subset lattice of (n) with sets ordered by inclusion. A collection I of subsets of (n) is called an ideal if every subset of a member of I is also in I. An intersecting family S in 2\sp{\lbrack n\rbrack } is called a star if there exists an element of (n) belonging to every member of S, and it is a 1-star if the intersection of every two members of I is exactly that element. Chvatal conjectured that if I is any ideal, then among the intersecting subfamilies of I of maximum cardinality there is a star. In Chapter 1, we prove Chvatal's conjecture for several special cases. Let I be an ideal in 2\sp{\lbrack n\rbrack } that is compressed with respect to a given element. We prove that among the largest intersecting families of I there is a star. We also prove that if the maximal elements B\sb1,\...,B\sb{q} of an ideal I can be partitioned into two 1-stars, then I satisfies Chvatal's conjecture.In Chapter 2, we consider the following two conjectures concerning intersecting families of a finite set. Conjecture 1: (Frankl and Furedi (18)) Given n, k, let ${\cal A}$ be a collection of subsets of an n-set such that 1 $\leq \vert A\cap B\vert \leq k$ for all A, B $\in {\cal A}$. Then \vert{\cal A}\vert \leq t\sb{n,k}, where t\sb{n,k} = \sum\sbsp{i=0}{k}{n-1\choose i}. Conjecture 2: (Snevily) Let S = \{ l\sb1, ...,l\sb{k}\} be a collection of k positive integers. If ${\cal A}$ is a collection of subsets of X such that $\vert A \cap B\vert \in S$ for all A, $B \in {\cal A}$, then \vert {\cal A}\vert \leq t\sb{n,k}. We prove that Conjecture 1 is true when n > 4.5k\sp{3} + 7.5k\sp2 + 3k + 1. We prove necessary conditions for possible counterexamples to Conjecture 2 when n is sufficiently large.Let ${\cal B}(k)$ denote the bipartite graph whose vertices are the k and k + 1 sets of (2k + 1), with edges specified by the inclusion relationship. Erdos conjectured that ${\cal B}(k)$ contains a Hamitonian cycle. Any such cycle must be composed of two matchings between the middle levels of the Boolean lattice. We study such matchings that are invariant under cyclic permutations of the ground set. We then construct a new class of matchings called modular matchings and show that these are nonisomorphic to the lexical matchings. We describe the orbits of the modular matchings under automorphisms of ${\cal B}(k),$ and we also construct an example of a matching that is neither lexical nor modular.Finally, we generalize some results about special vertex labelings that, by a theorem of Rosa, yield decompositions of the complete graph K\sb{n} into isomorphic copies of certain specified graphs.U of I OnlyETDs are only available to UIUC Users without author permissio

On pebbling graphs

The pebbling number of a graph G, f(G), is the least m such that, however m pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. We give another proof that f(Q^n) = 2^n (Chung) and show that for most graphs f(G) = |V(G)| or |V(G)| + 1. We also find explicitly for certain classes of graphs (i.e. for odd cycles and squares of paths). characterize efficient graphs, show that most graphs have the 2-pebbling property, and obtain some results on optimal pebbling