Induced Ramsey-type results and binary predicates for point sets

Let $k$ and $p$ be positive integers and let $Q$ be a finite point set in general position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is a finite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$ there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order type and $\binom{Q'}{p}$ is monochromatic in $c$. Ne\v{s}et\v{r}il and Valtr proved that for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They also proved that for every $k \ge 2$ and $p \ge 2$, there are point sets that are not $(k,p)$-Ramsey. As our main result, we introduce a new family of $(k,2)$-Ramsey point sets, extending a result of Ne\v{s}et\v{r}il and Valtr. We then use this new result to show that for every $k$ there is a point set $P$ such that no function $\Gamma$ that maps ordered pairs of distinct points from $P$ to a set of size $k$ can satisfy the following "local consistency" property: if $\Gamma$ attains the same values on two ordered triples of points from $P$, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.Comment: 22 pages, 3 figures, final version, minor correction

Ramsey numbers of ordered graphs

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of Combinatoric

Refinement of tissue diagnostics in the transplant program

Refinement of tissue diagnostics in the transplant program Abstract The thesis describes tissue diagnostics of organ transplants of the donor as well as recipient. In case of donated organs (grafts), it aims to improve the provisional scoring system of graft rejection after uterine transplantation. Using histopathological methods including immunohistochemistry, it compares lymphocytic population classified in the scoring system with histopathological findings within the resected uteri of women who underwent hysterectomy for benign reasons. The aim was to validate the necessity of the category of so called borderline changes, whose morphology was proven to overlap with physiological finding within healthy uterine cervix. Therefore, there is no need for immunosuppressive therapy in such cases. In case of recipients, the thesis investigates the role of pulmonary megakaryocytes in explanted lungs after transplantation. The idea was to elucidate pathogenesis of the diseases leading to the need of transplantation itself. Pulmonary megakaryocytes within explanted lungs of various recipients were mapped using histopathological methods and immunohistochemistry. The strikingly increased number of such cells was detected among vascular disorders, especially idiopathic pulmonary hypertension. This finding could...Zpřesnění tkáňové diagnostiky v transplantačním programu Abstrakt V této práci se zabýváme tkáňovou diagnostikou transplantovaných orgánů pocházejících od dárců i příjemců. U darovaných orgánů (graftů) jsme se zaměřili na upřesnění dosud provizorního skórovacího systému rejekčních změn v programu transplantace dělohy. Pomocí histopatologické diagnostiky, včetně imunohistochemických metod, jsme porovnali nálezy lymfocytárních infiltrátů uváděných ve zmíněném skórovacím systému s nálezy v resekátech děloh žen po hysterektomii provedené z důvodů nesouvisejících s děložním hrdlem. Cílem naší práce bylo ověřit platnost tzv. kategorie borderline změn - morfologický obraz těchto změn se však ukázal být překryvným s fyziologickým nálezem ve zdravém děložním hrdle a tudíž nevyžadujícím imunosupresivní terapii. U orgánů příjemců jsme se zaměřili na roli plicních megakaryocytů v explantátech po transplantaci plic. Cílem bylo přispět k ozřejmění patogeneze chorob vedoucích k nutnosti této transplantace. Plicní megakaryocyty jsme histopatologicky a imunohistochemicky mapovali ve vzorcích plicních explantátů příjemců s různými základními plicními chorobami a objevili jejich výrazně zvýšenou účast u vaskulárních plicních lézí. U idiopatické plicní hypertenze by tento nález mohl pomoci vysvětlit dosud kryptogenní cévní...Ústav patologie a molekulární medicínyDepartment of Pathology and Molecular MedicineSecond Faculty of Medicine2. lékařská fakult

Big Ramsey degrees of 3-uniform hypergraphs

Given a countably infinite hypergraph $\mathcal R$ and a finite hypergraph $\mathcal A$, the big Ramsey degree of $\mathcal A$ in $\mathcal R$ is the least number $L$ such that, for every finite $k$ and every $k$-colouring of the embeddings of $\mathcal A$ to $\mathcal R$, there exists an embedding $f$ from $\mathcal R$ to $\mathcal R$ such that all the embeddings of $\mathcal A$ to the image $f(\mathcal R)$ have at most $L$ different colours. We describe the big Ramsey degrees of the random countably infinite 3-uniform hypergraph, thereby solving a question of Sauer. We also give a new presentation of the results of Devlin and Sauer on, respectively, big Ramsey degrees of the order of the rationals and the countably infinite random graph. Our techniques generalise (in a natural way) to relational structures and give new examples of Ramsey structures (a concept recently introduced by Zucker with applications to topological dynamics).Comment: 8 pages, 3 figures, extended abstract for Eurocomb 201

Exact big Ramsey degrees via coding trees

We characterize the big Ramsey degrees of free amalgamation classes in finite binary languages defined by finitely many forbidden irreducible substructures, thus refining the recent upper bounds given by Zucker. Using this characterization, we show that the Fra\"iss\'e limit of each such class admits a big Ramsey structure. Consequently, the automorphism group of each such Fra\"iss\'e limit has a metrizable universal completion flow.Comment: Submitted versio

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

A Superlinear Lower Bound on the Number of 5-Holes

Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let h_5(n) be the minimum number of 5-holes among all sets of n points in the plane in general position. Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h_5(n) have been of order Omega(n) and O(n^2), respectively. We show that h_5(n) = Omega(n(log n)^(4/5)), obtaining the first superlinear lower bound on h_5(n). The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line l into two subsets, each of size at least 5 and not in convex position, then l intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted