# Two variants of the size Ramsey number

• Volume: 25, Issue: 1-2, page 141-149
• ISSN: 2083-5892

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## Abstract

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Given a graph H and an integer r ≥ 2, let G → (H,r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let $m\left(G\right)=ma{x}_{F\subseteq G}|E\left(F\right)|/|V\left(F\right)|$ and define the Ramsey density ${m}_{inf}\left(H,r\right)$ as the infimum of m(G) over all graphs G such that G → (H,r). In the first part of this paper we show that when H is a complete graph Kₖ on k vertices, then ${m}_{inf}\left(H,r\right)=\left(R-1\right)/2$, where R = R(k;r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited to Chvatál that the size Ramsey number for Kₖ equals $\left(\genfrac{}{}{0pt}{}{R}{2}\right)$. We also study an on-line version of the size Ramsey number, related to the following two-person game: Painter colors on-line the edges provided by Builder, and Painter’s goal is to avoid a monochromatic copy of Kₖ. The on-line Ramsey number R̅(k;r) is the smallest number of moves (edges) in which Builder can force Painter to lose if r colors are available. We show that R̅(3;2) = 8 and $R̅\left(k;2\right)\le 2k\left(\genfrac{}{}{0pt}{}{2k-2}{k-1}\right)$, but leave unanswered the question if R̅(k;2) = o(R²(k;2)).

## How to cite

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Andrzej Kurek, and Andrzej Ruciński. "Two variants of the size Ramsey number." Discussiones Mathematicae Graph Theory 25.1-2 (2005): 141-149. <http://eudml.org/doc/270499>.

@article{AndrzejKurek2005,
abstract = {Given a graph H and an integer r ≥ 2, let G → (H,r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let $m(G) = max_\{F ⊆ G\}|E(F)|/|V(F)|$ and define the Ramsey density $m_\{inf\}(H,r)$ as the infimum of m(G) over all graphs G such that G → (H,r). In the first part of this paper we show that when H is a complete graph Kₖ on k vertices, then $m_\{inf\}(H,r) = (R-1)/2$, where R = R(k;r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited to Chvatál that the size Ramsey number for Kₖ equals $\binom\{R\}\{2\}$. We also study an on-line version of the size Ramsey number, related to the following two-person game: Painter colors on-line the edges provided by Builder, and Painter’s goal is to avoid a monochromatic copy of Kₖ. The on-line Ramsey number R̅(k;r) is the smallest number of moves (edges) in which Builder can force Painter to lose if r colors are available. We show that R̅(3;2) = 8 and $R̅(k;2) ≤ 2k\binom\{2k-2\}\{k-1\}$, but leave unanswered the question if R̅(k;2) = o(R²(k;2)).},
author = {Andrzej Kurek, Andrzej Ruciński},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {size Ramsey number; graph density; online Ramsey games},
language = {eng},
number = {1-2},
pages = {141-149},
title = {Two variants of the size Ramsey number},
url = {http://eudml.org/doc/270499},
volume = {25},
year = {2005},
}

TY - JOUR
AU - Andrzej Kurek
AU - Andrzej Ruciński
TI - Two variants of the size Ramsey number
JO - Discussiones Mathematicae Graph Theory
PY - 2005
VL - 25
IS - 1-2
SP - 141
EP - 149
AB - Given a graph H and an integer r ≥ 2, let G → (H,r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let $m(G) = max_{F ⊆ G}|E(F)|/|V(F)|$ and define the Ramsey density $m_{inf}(H,r)$ as the infimum of m(G) over all graphs G such that G → (H,r). In the first part of this paper we show that when H is a complete graph Kₖ on k vertices, then $m_{inf}(H,r) = (R-1)/2$, where R = R(k;r) is the classical Ramsey number. As a corollary we derive a new proof of the result credited to Chvatál that the size Ramsey number for Kₖ equals $\binom{R}{2}$. We also study an on-line version of the size Ramsey number, related to the following two-person game: Painter colors on-line the edges provided by Builder, and Painter’s goal is to avoid a monochromatic copy of Kₖ. The on-line Ramsey number R̅(k;r) is the smallest number of moves (edges) in which Builder can force Painter to lose if r colors are available. We show that R̅(3;2) = 8 and $R̅(k;2) ≤ 2k\binom{2k-2}{k-1}$, but leave unanswered the question if R̅(k;2) = o(R²(k;2)).
LA - eng
KW - size Ramsey number; graph density; online Ramsey games
UR - http://eudml.org/doc/270499
ER -

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