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arXiv · 1911.06642

Generalized rainbow Tur\'an problems

Dániel Gerbner·Tamás Mészáros·Abhishek Methuku·Cory Palmer
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Abstract

Alon and Shikhelman initiated the systematic study of the following generalized Tur\'an problem: for fixed graphs $H$ and $F$ and an integer $n$, what is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph? An edge-colored graph is called rainbow if all its edges have different colors. The rainbow Tur\'an number of $F$ is defined as the maximum number of edges in a properly edge-colored graph on $n$ vertices with no rainbow copy of $F$. The study of rainbow Tur\'an problems was initiated by Keevash, Mubayi, Sudakov and Verstra\"ete. Motivated by the above problems, we study the following problem: What is the maximum number of copies of $F$ in a properly edge-colored graph on $n$ vertices without a rainbow copy of $F$? We establish several results, including when $F$ is a path, cycle or tree.

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Dániel Gerbner, Tamás Mészáros, Abhishek Methuku, Cory Palmer. 2019-11-15. Generalized rainbow Tur\'an problems. https://arxiv.org/abs/1911.06642

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