153 research outputs found
Colouring versus density in integers and Hales-Jewett cubes
We construct for every integer and every real a set of integers which, when coloured with
finitely many colours, contains a monochromatic -term arithmetic
progression, whilst every finite has a subset of
size that is free of arithmetic progressions of length .
This answers a question of Erd\H{o}s, Ne\v{s}et\v{r}il, and the second author.
Moreover, we obtain an analogous multidimensional statement and a Hales-Jewett
version of this result.Comment: 5 figure
Covering Partial Cubes with Zones
A partial cube is a graph having an isometric embedding in a hypercube.
Partial cubes are characterized by a natural equivalence relation on the edges,
whose classes are called zones. The number of zones determines the minimal
dimension of a hypercube in which the graph can be embedded. We consider the
problem of covering the vertices of a partial cube with the minimum number of
zones. The problem admits several special cases, among which are the problem of
covering the cells of a line arrangement with a minimum number of lines, and
the problem of finding a minimum-size fibre in a bipartite poset. For several
such special cases, we give upper and lower bounds on the minimum size of a
covering by zones. We also consider the computational complexity of those
problems, and establish some hardness results
Ramseyova teorie a kombinatorické hry
Ramsey theory studies the internal homogenity of mathematical structures (graphs, number sets), parts of which (subgraphs, number subsets) are arbitrarily coloured. Often, the sufficient object size implies the existence of a monochromatic sub-object. Combinatorial games are 2-player games of skill with perfect information. The theory of combinatorial games studies mostly the questions of existence of winning or drawing strategies. Let us consider an object that is studied by a particular Ramsey-type theorem. Assume two players alternately colour parts of this object by two colours and their goal is to create certain monochromatic sub-object. Then this is a combinatorial game. We focus on the minimum object size such that the appropriate Ramsey-type theorem holds, called Ramsey number, and on the minimum object size such that the rst player has a winning strategy in the corresponding combinatorial game, called game number. In this thesis, we describe such Ramsey-type theorems where the Ramsey number is substantially greater than the game number. This means, we show the existence of rst player's winning strategies, zogether with Ramsey and game numbers upper bounds, and we compare both numbers.Ramseyova teorie studuje vnitřní homogenitu matematických struktur (grafů, číselných oborů), jejichž části (podgrafy, podmnožiny) jsou libovolně obarveny. Často platí, že je-li studovaný objekt dostatečně velký, lze v něm najít určitý jednobarevný podobjekt. Kombinatorické hry jsou hry dvou hráčů s plnou informací, kde záleží pouze na jejich inteligenci. Teorie kombinatorických her studuje především otázky existence vyhrávajících či neprohrávajících strategií. Vezmeme-li ramseyovskou větu a necháme-li objekt, který tato věta studuje, střídavě barvit dvěma hráči, jejichž cílem je vytvořit určitý monochromatický podobjekt, dostaneme kombinatorickou hru. Předmětem našeho zájmu je jednak nejmenší velikost objektu, při které platí ramseyovská věta, tzv. ramseyovské číslo, a jednak nejmeněí velikost téhož objektu, při které má první hráč vyhrávající strategii v příslušné kombinatorické hře, tzv. herní číslo. V této práci popisujeme takové ramseyovské věty, u nichž je ramseyovské číslo podstatně větší než číslo herní. To znamená, že podáváme důkazy existence vyhrávajících strategií prvního hráče spolu s horními odhady na ramseyovská a herní čísla a obě čísla porovnáváme.Katedra aplikované matematikyDepartment of Applied MathematicsMatematicko-fyzikální fakultaFaculty of Mathematics and Physic
Ramsey theorem for trees with successor operation
We prove a general Ramsey theorem for trees with a successor operation. This
theorem is a common generalization of the Carlson-Simpson Theorem and the
Milliken Tree Theorem for regularly branching trees.
Our theorem has a number of applications both in finite and infinite
combinatorics. For example, we give a short proof of the unrestricted
Ne\v{s}et\v{r}il-R\"odl theorem, and we recover the Graham-Rothschild theorem.
Our original motivation came from the study of big Ramsey degrees - various
trees used in the study can be viewed as trees with a successor operation. To
illustrate this, we give a non-forcing proof of a theorem of Zucker on big
Ramsey degrees.Comment: 37 pages, 9 figure
Ergodic theorems for polynomials in nilpotent groups
The principal results proved in this expository thesis are the IP polynomial
Szemer\'edi theorem for nilpotent groups and the multiple term return times
theorem with nilsequence weights. It also contains extensions of the
convergence theorem for nilpotent polynomial multiple ergodic averages and the
return times theorem to locally compact second countable amenable groups.Comment: PhD thesis, University of Amsterdam, xi+146 pages. Based on
arXiv:1111.7292, arXiv:1206.0287, arXiv:1208.3977, arXiv:1210.5202,
arXiv:1301.188
Recent developments in graph Ramsey theory
Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress
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Problems and results on linear hypergraphs
In this thesis, we tackle several problems involving the study of 3-uniform, linear hypergraphs satisfying some additional structural constraint.
We begin with a problem of Hrushovski concerning Latin squares satisfying a partial associativity condition. From an Latin square one can define a binary operation , and is associative if and only if is a group multiplication table. Hrushovski asked whether, if is only associative a positive proportion of the time, must still in some sense be close to a group multiplication table. This problem manifests a well-studied combinatorial theme, in which a local structural constraint is relaxed (first to a `99' version and then to a `1' version) and the global consequences of the relaxed constraints are analysed. We show that the partial associativity condition is sufficient to deduce powerful global information, allowing us to find within a large subset with group-like structure. Since Latin squares can be regarded as 3-uniform, linear hypergraphs, and the partial associativity condition can be formulated in terms of the count of a particular subhypergraph, we are able to apply purely combinatorial methods to a problem that touches algebra, model theory and geometric group theory.
We then take this problem further. A condition due to Thomsen provides a combinatorial constraint which, if satisfied by the Latin square , proves that is in fact the multiplication table of an abelian group. It is then natural to ask whether a relaxed version of this result is also attainable, and by extending our methods we are able to prove a result of this flavour. Since the combinatorial obstructions to commutativity of are far more complex than those for associativity, topological complications arise that are not present in the earlier work.
We also study a problem of Loh concerning sequences of triples of integers from satisfying a certain `increasing' property. Loh studied the maximum length of such a sequence, improving a trivial upper bound of to using the triangle removal lemma and conjecturing that a natural construction of length is best possible. We provide the first power-type improvement to the upper bound, showing that there exists such that the length is bounded by . By viewing the triples as edges in a 3-uniform hypergraph, the increasing property shows that the hypergraph is linear and provides further restrictions in terms of forbidden subhypergraphs. By considering this formulation, we provide links to various important open problems including the Brown--Erd\H os--S\'os conjecture.
Finally, we present a collection of shorter results. In work connecting to the earlier chapters, we resolve the Brown--Erd\H os--S\'os conjecture in the context of hypergraphs with a group structure, and show moreover that subsets of group multiplication tables exhibit local density far beyond what can be hoped for in general. In work less closely connected to the main theme of the thesis, we also answer a question of Leader, Mili\'cevi\'c and Tan concerning partitions of boxes, consider a problem on projective cubes in , and resolve a conjecture concerning a diffusion process on graphs
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