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Full-Text Articles in Mathematics
A Machine Learning Approach To Constructing Ramsey Graphs Leads To The Trahtenbrot-Zykov Problem., Emily Hawboldt
A Machine Learning Approach To Constructing Ramsey Graphs Leads To The Trahtenbrot-Zykov Problem., Emily Hawboldt
Electronic Theses and Dissertations
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 …
The On-Line Width Of Various Classes Of Posets., Israel R. Curbelo
The On-Line Width Of Various Classes Of Posets., Israel R. Curbelo
Electronic Theses and Dissertations
An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains. Over 30 years ago, Szemer\'edi proved that any on-line algorithm could be forced to use $\binom{w+1}{2}$ chains to partition a poset of width $w$. The maximum number of chains that can be forced on any on-line algorithm remains unknown. In the survey paper by Bosek et al., variants of the problem were studied where the class is restricted to posets of bounded dimension or where the poset is presented via a realizer of size $d$. We prove …
Decisive Neutrality, Restricted Decisive Neutrality, And Split Decisive Neutrality On Median Semilattices And Median Graphs., Ulf Högnäs
Electronic Theses and Dissertations
Consensus functions on finite median semilattices and finite median graphs are studied from an axiomatic point of view. We start with a new axiomatic characterization of majority rule on a large class of median semilattices we call sufficient. A key axiom in this result is the restricted decisive neutrality condition. This condition is a restricted version of the more well-known axiom of decisive neutrality given in [4]. Our theorem is an extension of the main result given in [7]. Another main result is a complete characterization of the class of consensus on a finite median semilattice that satisfies the axioms …