Coloring number and on-line Ramsey theory for graphs and hypergraphs

Henry Kierstead, Goran Konjevod

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

Let c,s,t be positive integers. The (c,s,t)-Ramsey game is played by Builder and Painter. Play begins with an s-uniform hypergraph G 0=(V,E 0), where E 0=∅ and V is determined by Builder. On the ith round Builder constructs a new edge e i (distinct from previous edges) and sets G i =(V,E i ), where E i =E i-1∪{e i }. Painter responds by coloring e i with one of c colors. Builder wins if Painter eventually creates a monochromatic copy of K s t , the complete s-uniform hypergraph on t vertices; otherwise Painter wins when she has colored all possible edges. We extend the definition of coloring number to hypergraphs so that χ(G)≤col(G) for any hypergraph G and then show that Builder can win (c,s,t)-Ramsey game while building a hypergraph with coloring number at most col(K s t ). An important step in the proof is the analysis of an auxiliary survival game played by Presenter and Chooser. The (p,s,t)-survival game begins with an s-uniform hypergraph H 0 = (V,∅) with an arbitrary finite number of vertices and no edges. Let H i-1=(V i-1,E i-1) be the hypergraph constructed in the first i - 1 rounds. On the i-th round Presenter plays by presenting a p-subset P i ⊆V i-1 and Chooser responds by choosing an s-subset X i ⊆P i . The vertices in P i - X i are discarded and the edge X i added to E i-1 to form E i . Presenter wins the survival game if H i contains a copy of K s t for some i. We show that for positive integers p,s,t with -astr-temp≤p, Presenter has a winning strategy.

Original languageEnglish (US)
Pages (from-to)49-64
Number of pages16
JournalCombinatorica
Volume29
Issue number1
DOIs
StatePublished - Jan 2009

Fingerprint

Ramsey Theory
Coloring
Hypergraph
Colouring
Uniform Hypergraph
Game
Graph in graph theory
Integer
Subset
Color
Distinct
Arbitrary

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

Cite this

Coloring number and on-line Ramsey theory for graphs and hypergraphs. / Kierstead, Henry; Konjevod, Goran.

In: Combinatorica, Vol. 29, No. 1, 01.2009, p. 49-64.

Research output: Contribution to journalArticle

@article{1ef49c8f650643118977a83e11df5ccb,
title = "Coloring number and on-line Ramsey theory for graphs and hypergraphs",
abstract = "Let c,s,t be positive integers. The (c,s,t)-Ramsey game is played by Builder and Painter. Play begins with an s-uniform hypergraph G 0=(V,E 0), where E 0=∅ and V is determined by Builder. On the ith round Builder constructs a new edge e i (distinct from previous edges) and sets G i =(V,E i ), where E i =E i-1∪{e i }. Painter responds by coloring e i with one of c colors. Builder wins if Painter eventually creates a monochromatic copy of K s t , the complete s-uniform hypergraph on t vertices; otherwise Painter wins when she has colored all possible edges. We extend the definition of coloring number to hypergraphs so that χ(G)≤col(G) for any hypergraph G and then show that Builder can win (c,s,t)-Ramsey game while building a hypergraph with coloring number at most col(K s t ). An important step in the proof is the analysis of an auxiliary survival game played by Presenter and Chooser. The (p,s,t)-survival game begins with an s-uniform hypergraph H 0 = (V,∅) with an arbitrary finite number of vertices and no edges. Let H i-1=(V i-1,E i-1) be the hypergraph constructed in the first i - 1 rounds. On the i-th round Presenter plays by presenting a p-subset P i ⊆V i-1 and Chooser responds by choosing an s-subset X i ⊆P i . The vertices in P i - X i are discarded and the edge X i added to E i-1 to form E i . Presenter wins the survival game if H i contains a copy of K s t for some i. We show that for positive integers p,s,t with -astr-temp≤p, Presenter has a winning strategy.",
author = "Henry Kierstead and Goran Konjevod",
year = "2009",
month = "1",
doi = "10.1007/s00493-009-2264-1",
language = "English (US)",
volume = "29",
pages = "49--64",
journal = "Combinatorica",
issn = "0209-9683",
publisher = "Janos Bolyai Mathematical Society",
number = "1",

}

TY - JOUR

T1 - Coloring number and on-line Ramsey theory for graphs and hypergraphs

AU - Kierstead, Henry

AU - Konjevod, Goran

PY - 2009/1

Y1 - 2009/1

N2 - Let c,s,t be positive integers. The (c,s,t)-Ramsey game is played by Builder and Painter. Play begins with an s-uniform hypergraph G 0=(V,E 0), where E 0=∅ and V is determined by Builder. On the ith round Builder constructs a new edge e i (distinct from previous edges) and sets G i =(V,E i ), where E i =E i-1∪{e i }. Painter responds by coloring e i with one of c colors. Builder wins if Painter eventually creates a monochromatic copy of K s t , the complete s-uniform hypergraph on t vertices; otherwise Painter wins when she has colored all possible edges. We extend the definition of coloring number to hypergraphs so that χ(G)≤col(G) for any hypergraph G and then show that Builder can win (c,s,t)-Ramsey game while building a hypergraph with coloring number at most col(K s t ). An important step in the proof is the analysis of an auxiliary survival game played by Presenter and Chooser. The (p,s,t)-survival game begins with an s-uniform hypergraph H 0 = (V,∅) with an arbitrary finite number of vertices and no edges. Let H i-1=(V i-1,E i-1) be the hypergraph constructed in the first i - 1 rounds. On the i-th round Presenter plays by presenting a p-subset P i ⊆V i-1 and Chooser responds by choosing an s-subset X i ⊆P i . The vertices in P i - X i are discarded and the edge X i added to E i-1 to form E i . Presenter wins the survival game if H i contains a copy of K s t for some i. We show that for positive integers p,s,t with -astr-temp≤p, Presenter has a winning strategy.

AB - Let c,s,t be positive integers. The (c,s,t)-Ramsey game is played by Builder and Painter. Play begins with an s-uniform hypergraph G 0=(V,E 0), where E 0=∅ and V is determined by Builder. On the ith round Builder constructs a new edge e i (distinct from previous edges) and sets G i =(V,E i ), where E i =E i-1∪{e i }. Painter responds by coloring e i with one of c colors. Builder wins if Painter eventually creates a monochromatic copy of K s t , the complete s-uniform hypergraph on t vertices; otherwise Painter wins when she has colored all possible edges. We extend the definition of coloring number to hypergraphs so that χ(G)≤col(G) for any hypergraph G and then show that Builder can win (c,s,t)-Ramsey game while building a hypergraph with coloring number at most col(K s t ). An important step in the proof is the analysis of an auxiliary survival game played by Presenter and Chooser. The (p,s,t)-survival game begins with an s-uniform hypergraph H 0 = (V,∅) with an arbitrary finite number of vertices and no edges. Let H i-1=(V i-1,E i-1) be the hypergraph constructed in the first i - 1 rounds. On the i-th round Presenter plays by presenting a p-subset P i ⊆V i-1 and Chooser responds by choosing an s-subset X i ⊆P i . The vertices in P i - X i are discarded and the edge X i added to E i-1 to form E i . Presenter wins the survival game if H i contains a copy of K s t for some i. We show that for positive integers p,s,t with -astr-temp≤p, Presenter has a winning strategy.

UR - http://www.scopus.com/inward/record.url?scp=65549158263&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=65549158263&partnerID=8YFLogxK

U2 - 10.1007/s00493-009-2264-1

DO - 10.1007/s00493-009-2264-1

M3 - Article

AN - SCOPUS:65549158263

VL - 29

SP - 49

EP - 64

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 1

ER -