On Ordered Ramsey Numbers of Tripartite 3-Uniform Hypergraphs

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F21%3A10436871" target="_blank" >RIV/00216208:11320/21:10436871 - isvavai.cz</a>

Result on the web

<a href="https://doi.org/10.1007/978-3-030-83823-2_23" target="_blank" >https://doi.org/10.1007/978-3-030-83823-2_23</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/978-3-030-83823-2_23" target="_blank" >10.1007/978-3-030-83823-2_23</a>

Result language

angličtina

Original language name

On Ordered Ramsey Numbers of Tripartite 3-Uniform Hypergraphs

Original language description

For an integer $k geq 2$, an emph{ordered $k$-uniform hypergraph} $mathcal{H}=(H,&lt;)$ is a $k$-uniform hypergraph $H$ together with a fixed linear ordering $&lt;$ of its vertex set. The emph{ordered Ramsey number} $overline{R}(mathcal{H},mathcal{G})$ of two ordered $k$-uniform hypergraphs $mathcal{H}$ and $mathcal{G}$ is the smallest $N in mathbb{N}$ such that every red-blue coloring of the hyperedges of the ordered complete $k$-uniform hypergraph $mathcal{K}^{(k)}_N$ on $N$ vertices contains a blue copy of $mathcal{H}$ or a red copy of $mathcal{G}$. The ordered Ramsey numbers are quite extensively studied for ordered graphs, but little is known about ordered hypergraphs of higher uniformity. We provide some of the first nontrivial estimates on ordered Ramsey numbers of ordered 3-uniform hypergraphs. In particular, we prove that for all $d,n in mathbb{N}$ and for every ordered $3$-uniform hypergraph $mathcal{H}$ on $n$ vertices with maximum degree $d$ and with interval chromatic number $3$ there is an $varepsilon=varepsilon(d)&gt;0$ such that $$overline{R}(mathcal{H},mathcal{H}) leq 2^{O(n^{2-varepsilon})}.$$ In fact, we prove this upper bound for the number $overline{R}(mathcal{G},mathcal{K}^{(3)}_3(n))$, where $mathcal{G}$ is an ordered 3-uniform hypergraph with $n$ vertices and maximum degree $d$ and $mathcal{K}^{(3)}_3(n)$ is the ordered complete tripartite hypergraph with consecutive color classes of size $n$. We show that this bound is not far from the truth by proving $overline{R}(mathcal{H},mathcal{K}^{(3)}_3(n)) geq 2^{Omega(nlog{n})}$ for some fixed ordered $3$-uniform hypergraph $mathcal{H}$.

Czech name

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Czech description

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Type

D - Article in proceedings

CEP classification

—

OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GJ18-13685Y" target="_blank" >GJ18-13685Y: Model thoery and extremal combinatorics</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2021

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Article name in the collection

Extended Abstracts EuroComb 2021

ISBN

978-3-030-83823-2

ISSN

2297-0215

e-ISSN

2297-024X

Number of pages

6

Pages from-to

142-147

Publisher name

Springer International Publishing

Place of publication

neuveden

Event location

Barcelona

Event date

Sep 6, 2021

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

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