### Abstract

Motivated by a problem of Gallai on (1-1)-transversals of 2-intervals, it was proved by the authors in 1969 that if the edges of a complete graph K are colored with red and blue (both colors can appear on an edge) so that there is no monochromatic induced C_{4} and C_{5} then the vertices of K can be partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani recently strengthened this by showing that it is enough to assume that there is no induced monochromatic C_{4} and there is no induced C_{5} in one of the colors. Here we strengthen this result further, showing that it is enough to assume that there is no monochromatic induced C_{4} and there is no K5 on which both color classes induce a C_{5}. We also answer a question of Kaiser and Rabinovich, giving an example of six 2-convex sets in the plane such that any three intersect but there is no (1-1)-transversal for them.

Original language | English |
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Journal | Electronic Journal of Combinatorics |

Volume | 23 |

Issue number | 3 |

Publication status | Published - Sep 2 2016 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Theory and Mathematics

### Cite this

*Electronic Journal of Combinatorics*,

*23*(3).