### Abstract

The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family (formula presented) (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph K_{n} with t colors. Another area is to find the minimum number of monochromatic members of (formula presented) that partition or cover the vertex set of every edge colored complete graph. Here we propose a problem that connects these areas: for a fixed positive integers s ≤ t, at least how many vertices can be covered by the vertices of no more than s monochromatic members of (formula presented) in every edge coloring of K_{n} with t colors. Several problems and conjectures are presented, among them a possible extension of a well-known result of Cockayne and Lorimer on monochromatic matchings for which we prove an initial step: every t-coloring of K_{n} contains a (t − 1)-colored matching of size k provided that (formula presented)

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

Journal | Graphs and Combinatorics |

Volume | 31 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

- Matchings
- Monochromatic coverings
- Ramsey problems

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Graphs and Combinatorics*,

*31*(1). https://doi.org/10.1007/s00373-013-1372-2

**Coverings by Few Monochromatic Pieces : A Transition Between Two Ramsey Problems.** / Gyárfás, A.; Sárközy, Gábor N.; Selkow, Stanley.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 31, no. 1. https://doi.org/10.1007/s00373-013-1372-2

}

TY - JOUR

T1 - Coverings by Few Monochromatic Pieces

T2 - A Transition Between Two Ramsey Problems

AU - Gyárfás, A.

AU - Sárközy, Gábor N.

AU - Selkow, Stanley

PY - 2015

Y1 - 2015

N2 - The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family (formula presented) (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph Kn with t colors. Another area is to find the minimum number of monochromatic members of (formula presented) that partition or cover the vertex set of every edge colored complete graph. Here we propose a problem that connects these areas: for a fixed positive integers s ≤ t, at least how many vertices can be covered by the vertices of no more than s monochromatic members of (formula presented) in every edge coloring of Kn with t colors. Several problems and conjectures are presented, among them a possible extension of a well-known result of Cockayne and Lorimer on monochromatic matchings for which we prove an initial step: every t-coloring of Kn contains a (t − 1)-colored matching of size k provided that (formula presented)

AB - The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family (formula presented) (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph Kn with t colors. Another area is to find the minimum number of monochromatic members of (formula presented) that partition or cover the vertex set of every edge colored complete graph. Here we propose a problem that connects these areas: for a fixed positive integers s ≤ t, at least how many vertices can be covered by the vertices of no more than s monochromatic members of (formula presented) in every edge coloring of Kn with t colors. Several problems and conjectures are presented, among them a possible extension of a well-known result of Cockayne and Lorimer on monochromatic matchings for which we prove an initial step: every t-coloring of Kn contains a (t − 1)-colored matching of size k provided that (formula presented)

KW - Matchings

KW - Monochromatic coverings

KW - Ramsey problems

UR - http://www.scopus.com/inward/record.url?scp=84943588623&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84943588623&partnerID=8YFLogxK

U2 - 10.1007/s00373-013-1372-2

DO - 10.1007/s00373-013-1372-2

M3 - Article

AN - SCOPUS:84943588623

VL - 31

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 1

ER -