## Abstract

A well-known result of Rödl and Ruciński states that for any graph H there exists a constant C such that if (Formula presented.), then the random graph G_{n, p} is a.a.s. H-Ramsey, that is, any 2-coloring of its edges contains a monochromatic copy of H. Aside from a few simple exceptions, the corresponding 0-statement also holds, that is, there exists c > 0 such that whenever (Formula presented.) the random graph G_{n, p} is a.a.s. not H-Ramsey. We show that near this threshold, even when G_{n, p} is not H-Ramsey, it is often extremely close to being H-Ramsey. More precisely, we prove that for any constant c > 0 and any strictly 2-balanced graph H, if (Formula presented.), then the random graph G_{n, p} a.a.s. has the property that every 2-edge-coloring without monochromatic copies of H cannot be extended to an H-free coloring after (Formula presented.) extra random edges are added. This generalizes a result by Friedgut, Kohayakawa, Rödl, Ruciński, and Tetali, who in 2002 proved the same statement for triangles, and addresses a question raised by those authors. We also extend a result of theirs on the three-color case and show that these theorems need not hold when H is not strictly 2-balanced.

Original language | English |
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Pages (from-to) | 940-957 |

Number of pages | 18 |

Journal | Random Structures and Algorithms |

Volume | 57 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2020 Dec 1 |

### Bibliographical note

Funding Information:This research was supported by the Deutsche Forschungsgemeinschaft (DFG) project 415310276; the German‐Israeli Foundation for Scientific Research and Development, GIF grant G‐1347‐304.6/2016 [S.D.]. The Dahlem Research School, DRS Fellowship Program and the Berlin Mathematics Research Center MATH+, project “learning hypergraphs” [T.M.]. The European Research Council (ERC) starting grant RanDM 676632 [D.C.] and ERC consolidator grant PEPCo 724903 [J.L.]. Open access funding enabled and organized by Projekt DEAL. Funding information

Publisher Copyright:

© 2020 The Authors. Random Structures & Algorithms published by Wiley Periodicals LLC.

## All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics