On Induced Online Ramsey Number of Paths, Cycles, and Trees

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F19%3A00333848" target="_blank" >RIV/68407700:21240/19:00333848 - isvavai.cz</a>

Result on the web

<a href="https://link.springer.com/chapter/10.1007/978-3-030-19955-5_6" target="_blank" >https://link.springer.com/chapter/10.1007/978-3-030-19955-5_6</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/978-3-030-19955-5_6" target="_blank" >10.1007/978-3-030-19955-5_6</a>

Result language

angličtina

Original language name

On Induced Online Ramsey Number of Paths, Cycles, and Trees

Original language description

An online Ramsey game is a game between Builder and Painter, alternating in turns. They are given a fixed graph $H$ and a an infinite set of independent vertices $G$. In each round Builder draws a new edge in $G$ and Painter colors it either red or blue. Builder wins if after some finite round there is a monochromatic copy of the graph $H$, otherwise Painter wins. The online Ramsey number $widetilde{r}(H)$ is the minimum number of rounds such that Builder can force a monochromatic copy of $H$ in $G$. This is an analogy to the size-Ramsey number $overline{r}(H)$ defined as the minimum number such that there exists graph $G$ with $overline{r}(H)$ edges where for any edge two-coloring $G$ contains a monochromatic copy of $H$. In this extended abstract, we introduce the concept of induced online Ramsey numbers: the induced online Ramsey number $overline{r}_{ind}(H)$ is the minimum number of rounds Builder can force an induced monochromatic copy of $H$ in $G$. We prove asymptotically tight bounds on the induced online Ramsey numbers of paths, cycles and two families of trees. Moreover, we provide a result analogous to Conlon [On-line Ramsey Numbers, SIAM J. Discr. Math. 2009], showing that there is an infinite family of trees $T_1,T_2,dots$, $|T_i|<|T_{i+1}|$ for $ige1$, such that [ lim_{itoinfty} frac{widetilde{r}(T_i)}{overline{r}(T_i)} = 0. ]

Czech name

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Czech description

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Type

D - Article in proceedings

CEP classification

—

OECD FORD branch

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Project

—

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2019

Confidentiality

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

Article name in the collection

The 14th International Computer Science Symposium in Russia

ISBN

978-3-030-19954-8

ISSN

0302-9743

e-ISSN

—

Number of pages

10

Pages from-to

60-69

Publisher name

Springer, Cham

Place of publication

—

Event location

Novosibirsk

Event date

Jul 1, 2019

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

000490894900006