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Semigroup-valued metric spaces
Konečný, Matěj ; Hubička, Jan (advisor)
The structural Ramsey theory is a field on the boundary of combinatorics and model theory with deep connections to topological dynamics. Most of the known Ramsey classes in finite binary symmetric relational language can be shown to be Ramsey by utilizing a variant of the shortest path completion (e.g. Sauer's S-metric spaces, Conant's generalised metric spaces, Braunfeld's Λ-ultrametric spaces or Cherlin's metrically homogeneous graphs). In this thesis we explore the limits of the shortest path completion. We offer a unifying framework - semigroup-valued metric spaces - for all the aforementioned Ramsey classes and study their Ramsey expansions and EPPA (the extension property for partial automorphisms). Our results can be seen as evidence for the importance of studying the completion problem for amalgamation classes and have some further applications (such as the stationary independence relation). As a corollary of our general theorems, we reprove results of Hubička and Nešetřil on Sauer's S-metric spaces, results of Hubička, Nešetřil and the author on Conant's generalised metric spaces, Braunfeld's results on Λ-ultrametric spaces and the results of Aranda et al. on Cherlin's primitive 3-constrained metrically homogeneous graphs. We also solve several open problems such as EPPA for Λ-ultrametric...
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Semigroup-valued metric spaces
Konečný, Matěj ; Hubička, Jan (advisor)
The structural Ramsey theory is a field on the boundary of combinatorics and model theory with deep connections to topological dynamics. Most of the known Ramsey classes in finite binary symmetric relational language can be shown to be Ramsey by utilizing a variant of the shortest path completion (e.g. Sauer's S-metric spaces, Conant's generalised metric spaces, Braunfeld's Λ-ultrametric spaces or Cherlin's metrically homogeneous graphs). In this thesis we explore the limits of the shortest path completion. We offer a unifying framework - semigroup-valued metric spaces - for all the aforementioned Ramsey classes and study their Ramsey expansions and EPPA (the extension property for partial automorphisms). Our results can be seen as evidence for the importance of studying the completion problem for amalgamation classes and have some further applications (such as the stationary independence relation). As a corollary of our general theorems, we reprove results of Hubička and Nešetřil on Sauer's S-metric spaces, results of Hubička, Nešetřil and the author on Conant's generalised metric spaces, Braunfeld's results on Λ-ultrametric spaces and the results of Aranda et al. on Cherlin's primitive 3-constrained metrically homogeneous graphs. We also solve several open problems such as EPPA for Λ-ultrametric...
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Computing and estimating ordered Ramsey numbers
Poljak, Marian ; Balko, Martin (advisor) ; Hubička, Jan (referee)
We study ordered Ramsey numbers, which are an analogue of the classical Ramsey numbers for ordered graphs. We improve some already obtained results for a special class of ordered matchings and disprove a conjecture of Rohatgi. We expand the classical notion of Ramsey goodness to the ordered case and we attempt to characterize all Ram- sey good connected ordered graphs. We outline how Ramsey numbers can be obtained computationally and describe our SAT solver based utility developed to achieve this goal, which might be of use to other researchers studying this topic. 1
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Semigroup-valued metric spaces
Konečný, Matěj ; Hubička, Jan (advisor) ; Pultr, Aleš (referee)
The structural Ramsey theory is a field on the boundary of combinatorics and model theory with deep connections to topological dynamics. Most of the known Ramsey classes in finite binary symmetric relational language can be shown to be Ramsey by utilizing a variant of the shortest path completion (e.g. Sauer's S-metric spaces, Conant's generalised metric spaces, Braunfeld's Λ-ultrametric spaces or Cherlin's metrically homogeneous graphs). In this thesis we explore the limits of the shortest path completion. We offer a unifying framework - semigroup-valued metric spaces - for all the aforementioned Ramsey classes and study their Ramsey expansions and EPPA (the extension property for partial automorphisms). Our results can be seen as evidence for the importance of studying the completion problem for amalgamation classes and have some further applications (such as the stationary independence relation). As a corollary of our general theorems, we reprove results of Hubička and Nešetřil on Sauer's S-metric spaces, results of Hubička, Nešetřil and the author on Conant's generalised metric spaces, Braunfeld's results on Λ-ultrametric spaces and the results of Aranda et al. on Cherlin's primitive 3-constrained metrically homogeneous graphs. We also solve several open problems such as EPPA for Λ-ultrametric...
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Combinatorial Properties of Metrically Homogeneous Graphs
Konečný, Matěj ; Hubička, Jan (advisor) ; Nešetřil, Jaroslav (referee)
Ramsey theory looks for regularities in large objects. Model theory studies algebraic structures as models of theories. The structural Ramsey theory com- bines these two fields and is concerned with Ramsey-type questions about certain model-theoretic structures. In 2005, Nešetřil initiated a systematic study of the so-called Ramsey classes of finite structures. This thesis is a contribution to the programme; we find Ramsey expansions of the primitive 3-constrained classes from Cherlin's catalogue of metrically homogeneous graphs. A key ingredient is an explicit combinatorial algorithm to fill-in the missing distances in edge-labelled graphs to obtain structures from Cherlin's classes. This algorithm also implies the extension property for partial automorphisms (EPPA), another combinatorial property of classes of finite structures. 1
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Combinatorial Games Theory
Valla, Tomáš ; Nešetřil, Jaroslav (advisor) ; Sgall, Jiří (referee) ; Spirakis, Paul (referee)
Title: Combinatorial Games Theory Author: Tomáš Valla Department / Institute: IUUK MFF UK Supervisor: Prof. RNDr. Jaroslav Nešetřil, DrSc., IUUK MFF UK Abstract: In this thesis we study the complexity that appears when we consider the competitive version of a certain environment or process, using mainly the tools of al- gorithmic game theory, complexity theory, and others. For example, in the Internet environment, one cannot apply any classical graph algorithm on the graph of connected computers, because it usually requires existence of a central authority, that manipu- lates with the graph. We describe a local and distributed game, that in a competitive environment without a central authority simulates the computation of the weighted vertex cover, together with generalisation to hitting set and submodular weight func- tion. We prove that this game always has a Nash equilibrium and each equilibrium yields the same approximation of optimal cover, that is achieved by the best known ap- proximation algorithms. More precisely, the Price of Anarchy of our game is the same as the best known approximation ratio for this problem. All previous results in this field do not have the Price of Anarchy bounded by a constant. Moreover, we include the results in two more fields, related to the complexity of competitive...
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Erdos-Szekeres type theorems
Eliáš, Marek ; Matoušek, Jiří (advisor) ; Cibulka, Josef (referee)
Let P = (p1, p2, . . . , pN ) be a sequence of points in the plane, where pi = (xi, yi) and x1 < x2 < · · · < xN . A famous 1935 Erdős-Szekeres theorem asserts that every such P contains a monotone subsequence S of √ N points. Another, equally famous theorem from the same paper implies that every such P contains a convex or concave subsequence of Ω(log N) points. First we define a (k + 1)-tuple K ⊆ P to be positive if it lies on the graph of a function whose kth derivative is everywhere nonnegative, and similarly for a negative (k + 1)-tuple. Then we say that S ⊆ P is kth-order monotone if its (k + 1)- tuples are all positive or all negative. In this thesis we investigate quantitative bound for the corresponding Ramsey-type result. We obtain an Ω(log(k−1) N) lower bound ((k − 1)-times iterated logarithm). We also improve bounds for related problems: Order types and One-sided sets of hyperplanes. 1
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