The rainbow Turan number $\mathrm{ex}^*(n,H)$ of a graph $H$ is the maximum possible number of edges in a properly edge-coloured $n$-vertex graph with no rainbow subgraph isomorphic to $H$. We prove that for any integer $k\geq 2$, $\mathrm{ex}^*(n,C_{2k})=O(n^{1+1/k})$. This is tight and establishes a conjecture of Keevash, Mubayi, Sudakov and Verstraete. We use the same method to prove several other conjectures in various topics. First, we prove that there exists a constant $c$ such that any properly edge-coloured $n$-vertex graph with more than $cn(\log n)^4$ edges contains a rainbow cycle. It is known that there exist properly edge-coloured $n$-vertex graphs with $\Omega(n\log n)$ edges which do not contain any rainbow cycle. Secondly, we show that in any proper edge-colouring of $K_n$ with $o(n^{\frac{r}{r-1}\cdot \frac{k-1}{k}})$ colours, there exist $r$ colour-isomorphic, pairwise vertex-disjoint copies of $C_{2k}$. This proves in a strong form a conjecture of Conlon and Tyomkyn, and a strenghtened version proposed by Xu, Zhang, Jing and Ge. Moreover, we answer a question of Jiang and Newman by showing that there exists a constant $c=c(r)$ such that any $n$-vertex graph with more than $cn^{2-1/r}(\log n)^{7/r}$ edges contains the $r$-blowup of an even cycle. Finally, we prove that the $r$-blowup of $C_{2k}$ has Turan number $O(n^{2-\frac{1}{r}+\frac{1}{k+r-1}+o(1)})$, which can be used to disprove an old conjecture of Erd\H os and Simonovits.

Rainbow Turán number of even cycles, repeated patterns and blow-ups of cycles

Considering a natural generalization of the Ruzsa–Szemeredi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing copies of Ks such that any Kr is contained in at most one Ks. We also give bounds for the generalized rainbow Turan problem ex (n, H, rainbow - F) when F is complete. In particular, we answer a question of Gerbner, Meszaros, Methuku and Palmer, showing that there are properly edge-coloured graphs on n vertices with copies of Kr such that no Kr is rainbow.

/pdf/generalizations-of-the-ruzsa-szemeredi-and-rainbow-turan-5eymwbxwz2.pdf

Generalizations of the Ruzsa–Szemerédi and rainbow Turán problems for cliques

Let $F$ be a fixed graph. The rainbow Turán number of $F$ is defined as the maximum number of edges in a graph on $n$ vertices that has a proper edge-coloring with no rainbow copy of $F$ (i.e., a copy of $F$ all of whose edges have different colours). The systematic study of such problems was initiated by Keevash, Mubayi, Sudakov and Verstraëte. \n In this paper, we show that the rainbow Turán number of a path with $k+1$ edges is less than $\\left(9k/7+2\\right) n$, improving an earlier estimate of Johnston,  Palmer and Sarkar.

/pdf/on-the-rainbow-turan-number-of-paths-26jhrs2xbo.pdf

On the Rainbow Turan number of paths

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n, F)$, is the rainbow Turan number of $F$. We show that $ex^*(n,P_k)\geq \frac{k}{2}n + O(1)$ where $P_k$ is a path on $k\geq 3$ edges, generalizing a result by Maamoun and Meyniel and by Johnston, Palmer and Sarkar. We show similar bounds for brooms on $2^s-1$ edges and diameter $\leq 10$ and a few other caterpillars of small diameter.

Lower bounds for rainbow Tur\'{a}n numbers of paths and other trees

Alon and Shikhelman initiated the systematic study of the following generalized Turan 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 Turan 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 Turan problems was initiated by Keevash, Mubayi, Sudakov and Verstraete. 
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.

Generalized rainbow Tur\'an problems

For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n,F)$, is the rainbow Tur a n number  of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstra e te [ Combinatorics, Probability and Computing 16 (2007)]. We determine $ex^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on $ex^*(n,F)$ when $F$ is a path with $l$ edges, disproving a conjecture in the aforementioned paper for $l=4$.

/pdf/rainbow-turan-problems-for-paths-and-forests-of-stars-1ze4bdxjhj.pdf

Rainbow Turán Problems for Paths and Forests of Stars

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Rainbow Turán Problems for Paths and Forests of Stars