### Abstract

In 1983, Chvátal, Trotter and the two senior authors proved that for any Δ there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph KN with N≥Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Δ. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and O(N^{2-1/Δ}log^{1/Δ}N) edges, with N=⌈B'n⌉ for some constant B.⌉ that depends only on Δ. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Δ is O(n2-1/Δlog1/Δn). Our approach is based on random graphs; in fact, we show that the classical Erdõs-Rényi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Δ.The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed.

Original language | English |
---|---|

Pages (from-to) | 5041-5065 |

Number of pages | 25 |

Journal | Advances in Mathematics |

Volume | 226 |

Issue number | 6 |

DOIs | |

Publication status | Published - Apr 1 2011 |

### Fingerprint

### Keywords

- Inheritance of regularity
- Random graphs
- Regularity lemma
- Size-Ramsey numbers
- Universal graphs

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*226*(6), 5041-5065. https://doi.org/10.1016/j.aim.2011.01.004

**Sparse partition universal graphs for graphs of bounded degree.** / Kohayakawa, Yoshiharu; Rödl, Vojtěch; Schacht, Mathias; Szemerédi, E.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 226, no. 6, pp. 5041-5065. https://doi.org/10.1016/j.aim.2011.01.004

}

TY - JOUR

T1 - Sparse partition universal graphs for graphs of bounded degree

AU - Kohayakawa, Yoshiharu

AU - Rödl, Vojtěch

AU - Schacht, Mathias

AU - Szemerédi, E.

PY - 2011/4/1

Y1 - 2011/4/1

N2 - In 1983, Chvátal, Trotter and the two senior authors proved that for any Δ there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph KN with N≥Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Δ. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and O(N2-1/Δlog1/ΔN) edges, with N=⌈B'n⌉ for some constant B.⌉ that depends only on Δ. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Δ is O(n2-1/Δlog1/Δn). Our approach is based on random graphs; in fact, we show that the classical Erdõs-Rényi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Δ.The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed.

AB - In 1983, Chvátal, Trotter and the two senior authors proved that for any Δ there exists a constant B such that, for any n, any 2-colouring of the edges of the complete graph KN with N≥Bn vertices yields a monochromatic copy of any graph H that has n vertices and maximum degree Δ. We prove that the complete graph may be replaced by a sparser graph G that has N vertices and O(N2-1/Δlog1/ΔN) edges, with N=⌈B'n⌉ for some constant B.⌉ that depends only on Δ. Consequently, the so-called size-Ramsey number of any H with n vertices and maximum degree Δ is O(n2-1/Δlog1/Δn). Our approach is based on random graphs; in fact, we show that the classical Erdõs-Rényi random graph with the numerical parameters above satisfies a stronger partition property with high probability, namely, that any 2-colouring of its edges contains a monochromatic universal graph for the class of graphs on n vertices and maximum degree Δ.The main tool in our proof is the regularity method, adapted to a suitable sparse setting. The novel ingredient developed here is an embedding strategy that allows one to embed bounded degree graphs of linear order in certain pseudorandom graphs. Crucial to our proof is the fact that regularity is typically inherited at a scale that is much finer than the scale at which it is assumed.

KW - Inheritance of regularity

KW - Random graphs

KW - Regularity lemma

KW - Size-Ramsey numbers

KW - Universal graphs

UR - http://www.scopus.com/inward/record.url?scp=79953839272&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79953839272&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2011.01.004

DO - 10.1016/j.aim.2011.01.004

M3 - Article

AN - SCOPUS:79953839272

VL - 226

SP - 5041

EP - 5065

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

IS - 6

ER -