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

The oriented size Ramsey number of directed paths

An oriented graph is a directed graph with no bi-directed edges, i.e. if xy is an edge then yx is not an edge. The oriented size Ramsey number of an oriented graph H, denoted by \vec{r}(H), is the minimum m for which there exists an oriented graph G with m edges, such that every 2-colouring of G contains a monochromatic copy of H. In this paper we prove that the oriented size Ramsey number of the directed paths on n vertices satisfies \vec{r}(\vec{P}_{n}) = \Omega (n^{2} log n). This improves a lower bound by Ben-Eliezer, Krivelevich and Sudakov. It also matches an upper bound by Bucić and the authors, thus establishing an asymptotically tight bound on \vec{r}(\vec{P}_{n}). We also discuss how our methods can be used to improve the best known lower bound of the k-colour version of \vec{r}(\vec{P}_{n})

Ramsey numbers of hypergraphs of a given size

The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erd\H{o}s and Graham asked to maximize the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed $k \ge 3$ and $q \ge 2$ we prove that the largest possible $q$-color Ramsey number of a $k$-uniform hypergraph with $m$ edges is at most $\mathrm{tw}_k(O(\sqrt{m})),$ where $\mathrm{tw}$ denotes the tower function. We also present a construction showing that this bound is tight for $q \ge 4$. This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for $k \geq 4$ and the lower bound for $k=3$. Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs

Kombinatorikus informatika = Combinatorial Computer Science

A [1, 3, 4, 7-9, 13, 14] eredmények közvetlenül kódokhoz kapcsolódnak. A [7, 9, 3] dolgozatokban bevezettük az egy felhasználót meghatározó superimposed kódokat, majd Alon és Körner eredményeit felhasználva sikerült kód korlátokat kapni. Vizsgáltuk az identifying kódókat véletlen gráfokban [1,8]. Itt (1 valószínüséggel) éles korlátokat kaptunk a minimális kód méretére. A kódokhoz kapcsolódó Turán rendszereket írtunk le a The CRC Handbook of Combinatorial Designs-ba [13]. Ezen kívül könyvet írunk a többszörös hozzáférésű csatornák kódolásáról, 250 oldal elkészült. A kódok korrelációs tulajdonságaihoz kapcsolódó véletlen metsző halmaz-rendszerek tulajdonságait (1 valószínüséggel) leírtuk adott részhalmaz méreten belül [5,6]. Az elméleti informatikában nagyon fontos Ramsey problémákat vizsgáltunk [2,10-12]. A Szemerédi lemma segítségével néhány régi Ramsey típusú nyitott problémát sikerült megoldani. Meghatároztuk az utak Ramsey számát pontosan három szín esetén (több, mint 25 éven át volt nyitott probléma), a körfedési számot pontosítottuk és teljes páros gráfok Ramsey szinezéseit is vizsgáltuk. | Our results in [1, 3, 4, 7-9, 13, 14] are related to codes. In [7, 9, 3] we introduced the single user tracing superimposed codes and using results of Alon, Körner we gave bounds on their minimum length. We investigated identifying codes in random graphs [1, 8]. We obtained (with probability 1) tight bounds on the minimum size of the code. We described the Turán systems related to codes [13]. We are writing a book on coding of multiple access channels, 250 pages are ready. The related to correlation properties of codes, random intersecting systems were described (with probability 1) in [5, 6] up to a certain subset size. We investigated some Ramsey problems [2,10-12] which in general are very important in theoretical computer science. Using the Szemerédi lemma we managed to solve some long standing open Ramsey problems. We determined exactly the Ramsey number of paths in case of three colors (this problem was open for more than 25 years), narrowed the bounds on cycle partition number and we also investigated the Ramsey colorings of bipartite graphs

The Ramsey Number for 3-Uniform Tight Hypergraph Cycles

Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .–.–., vn and edges v1v2v3, v2v3v4, .–.–., vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl