### Abstract

Alon, Balogh, Keevash, and Sudakov proved that the (κ - 1)-partite Turán graph maximizes the number of distinct r-edge-colorings with no monochromatic Kκ for all fixed κ and r = 2, 3, among all n-vertex graphs. In this paper, we determine this function asymptotically for r = 2 among n-vertex graphs with a sublinear independence number. Somewhat surprisingly, unlike Alon, Balog, Keevash, and Sudakov'fs result, the extremal construction from Ramsey.Tura7acute;n theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with a sublinear independence number, even in the 2-colored case. In the second problem, we determine the maximum number of triangles asymptotically in an n-vertex Kκ free graph G with α(G) = o(n). The extremal graphs have a similar structure to the extremal graphs for the classical Ramsey.Turán problem, i.e., when the number of edges is maximized.

Original language | English (US) |
---|---|

Pages (from-to) | 1848-1866 |

Number of pages | 19 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 31 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2017 |

### Fingerprint

### Keywords

- Edge-coloring
- Monochromatic cliques
- Ramsey-Turán

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*31*(3), 1848-1866. https://doi.org/10.1137/16M1086078

**On two problems in ramsey-turán theory.** / Balogh, József; Liu, Hong; Sharifzadeh, Maryam.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 31, no. 3, pp. 1848-1866. https://doi.org/10.1137/16M1086078

}

TY - JOUR

T1 - On two problems in ramsey-turán theory

AU - Balogh, József

AU - Liu, Hong

AU - Sharifzadeh, Maryam

PY - 2017/1/1

Y1 - 2017/1/1

N2 - Alon, Balogh, Keevash, and Sudakov proved that the (κ - 1)-partite Turán graph maximizes the number of distinct r-edge-colorings with no monochromatic Kκ for all fixed κ and r = 2, 3, among all n-vertex graphs. In this paper, we determine this function asymptotically for r = 2 among n-vertex graphs with a sublinear independence number. Somewhat surprisingly, unlike Alon, Balog, Keevash, and Sudakov'fs result, the extremal construction from Ramsey.Tura7acute;n theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with a sublinear independence number, even in the 2-colored case. In the second problem, we determine the maximum number of triangles asymptotically in an n-vertex Kκ free graph G with α(G) = o(n). The extremal graphs have a similar structure to the extremal graphs for the classical Ramsey.Turán problem, i.e., when the number of edges is maximized.

AB - Alon, Balogh, Keevash, and Sudakov proved that the (κ - 1)-partite Turán graph maximizes the number of distinct r-edge-colorings with no monochromatic Kκ for all fixed κ and r = 2, 3, among all n-vertex graphs. In this paper, we determine this function asymptotically for r = 2 among n-vertex graphs with a sublinear independence number. Somewhat surprisingly, unlike Alon, Balog, Keevash, and Sudakov'fs result, the extremal construction from Ramsey.Tura7acute;n theory, as a natural candidate, does not maximize the number of distinct edge-colorings with no monochromatic cliques among all graphs with a sublinear independence number, even in the 2-colored case. In the second problem, we determine the maximum number of triangles asymptotically in an n-vertex Kκ free graph G with α(G) = o(n). The extremal graphs have a similar structure to the extremal graphs for the classical Ramsey.Turán problem, i.e., when the number of edges is maximized.

KW - Edge-coloring

KW - Monochromatic cliques

KW - Ramsey-Turán

UR - http://www.scopus.com/inward/record.url?scp=85031712530&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85031712530&partnerID=8YFLogxK

U2 - 10.1137/16M1086078

DO - 10.1137/16M1086078

M3 - Article

AN - SCOPUS:85031712530

VL - 31

SP - 1848

EP - 1866

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 3

ER -