Computers in Ramsey Theory; Testing, Constructions and Nonexistence

Computers in Ramsey Theory Ramsey theory is often regarded as the study of how order emerges from randomness. While originated in mathematical logic, it has applications in geometry, number theory, game theory, information theory, approximation algorithms, and other areas of mathematics and theoretical computer science. Ramsey theory studies the conditions of when a combinatorial object necessarily contains some smaller given objects. The central concept in Ramsey theory is that of arrowing, which in the case of graphs describes when colorings of larger graphs necessarily contain monochromatic copies of given smaller graphs. The role of Ramsey numbers is to quantify some of the general existential theorems in Ramsey theory, always involving arrowing. The determination of whether this arrowing holds is notoriously difficult, and thus it leads to numerous computational challenges concerning various types of Ramsey numbers and closely related Folkman numbers. This talk will overview how computers are increasingly used to study the bounds on Ramsey and Folkman numbers, and properties of Ramsey arrowing in general. This is happening in the area where traditional approaches typically call for classical computer-free proofs. It is evident that now we understand Ramsey theory much better than a few decades ago, increasingly due to computations. Further, more such progress and new insights based on computations should be anticipated

Ramsey properties of products and pullbacks of categories and the Grothendieck construction

In this paper we provide purely categorical proofs of two important results of structural Ramsey theory: the result of M.\ Soki\'c that the free product of Ramsey classes is a Ramsey class and the result of M.\ Bodirsky that adding constants to the language of a Ramsey class preserves the Ramsey property. The proofs that we present here ignore the model-theoretic background of these statements. Instead, they focus on categorical constructions by which the classes can be constructed, generalizing the original statements along the way. It turns out that the restriction to classes of relational structures, although fundamental for the original proof strategies, is not relevant for the statements themselves. The categorical proofs we present here remove all the restrictions on the signature of first-order structures and provide the information not only about the Ramsey property but also about the Ramsey degrees

Effective Techniques in Reverse Mathematics

We develop the theory of strong reductions in the reverse mathematics zoo and show by means of novel tree labeling constructions that strong computable reduction is sufficient to separate four stable relatives of Ramsey\u27s theorem for 2-colorings of pairs

A machine learning approach to constructing Ramsey graphs leads to the Trahtenbrot-Zykov problem.

Attempts at approaching the well-known and difficult problem of constructing Ramsey graphs via machine learning lead to another difficult problem posed by Zykov in 1963 (now commonly referred to as the Trahtenbrot-Zykov problem): For which graphs F does there exist some graph G such that the neighborhood of every vertex in G induces a subgraph isomorphic to F? Chapter 1 provides a brief introduction to graph theory. Chapter 2 introduces Ramsey theory for graphs. Chapter 3 details a reinforcement learning implementation for Ramsey graph construction. The implementation is based on board game software, specifically the AlphaZero program and its success learning to play games from scratch. The chapter ends with a description of how computing challenges naturally shifted the project towards the Trahtenbrot-Zykov problem. Chapter 3 also includes recommendations for continuing the project and attempting to overcome these challenges. Chapter 4 defines the Trahtenbrot-Zykov problem and outlines its history, including proofs of results omitted from their original papers. This chapter also contains a program for constructing graphs with all neighborhood-induced subgraphs isomorphic to a given graph F. The end of Chapter 4 presents constructions from the program when F is a Ramsey graph. Constructing such graphs is a non-trivial task, as Bulitko proved in 1973 that the Trahtenbrot-Zykov problem is undecidable. Chapter 5 is a translation from Russian to English of this famous result, a proof not previously available in English. Chapter 6 introduces Cayley graphs and their relationship to the Trahtenbrot-Zykov problem. The chapter ends with constructions of Cayley graphs Γ in which the neighborhood of every vertex of Γ induces a subgraph isomorphic to a given Ramsey graph, which leads to a conjecture regarding the unique extremal Ramsey(4, 4) graph

State of B\"uchi Complementation

Complementation of B\"uchi automata has been studied for over five decades since the formalism was introduced in 1960. Known complementation constructions can be classified into Ramsey-based, determinization-based, rank-based, and slice-based approaches. Regarding the performance of these approaches, there have been several complexity analyses but very few experimental results. What especially lacks is a comparative experiment on all of the four approaches to see how they perform in practice. In this paper, we review the four approaches, propose several optimization heuristics, and perform comparative experimentation on four representative constructions that are considered the most efficient in each approach. The experimental results show that (1) the determinization-based Safra-Piterman construction outperforms the other three in producing smaller complements and finishing more tasks in the allocated time and (2) the proposed heuristics substantially improve the Safra-Piterman and the slice-based constructions.Comment: 28 pages, 4 figures, a preliminary version of this paper appeared in the Proceedings of the 15th International Conference on Implementation and Application of Automata (CIAA