### 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) |
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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 |

### Keywords

- Edge-coloring
- Monochromatic cliques
- Ramsey-Turán

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*SIAM Journal on Discrete Mathematics*,

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