Rainbow Turán Problems

For a fixed graph H, we define the rainbow Turán number ex^*(n,H) to be the maximum number of edges in a graph on n vertices that has a proper edge-colouring with no rainbow H. Recall that the (ordinary) Turán number ex(n,H) is the maximum number of edges in a graph on n vertices that does not contain a copy of H. For any non-bipartite H we show that ex^*(n,H)=(1+o(1))ex(n,H), and if H is colour-critical we show that ex^{*}(n,H)=ex(n,H). When H is the complete bipartite graph K_{s,t} with s ≤ t we show ex^*(n,K_{s,t}) = O(n^{2-1/s}), which matches the known bounds for ex(n,K_{s,t}) up to a constant. We also study the rainbow Turán problem for even cycles, and in particular prove the bound ex^*(n,C_6) = O(n^{4/3}), which is of the correct order of magnitude

Graphs without a rainbow path of length 3

In 1959 Erd\H{o}s and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. Here we study a rainbow version of their theorem, in which one considers $k \geq 1$ graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximal number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any $k \geq 1$

Essentially tight bounds for rainbow cycles in proper edge-colourings

An edge-coloured graph is said to be rainbow if no colour appears more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstra\"ete from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on $n$ vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of $(\log n)^{2+o(1)}$ for this question. Very recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the $o(1)$ term in Tomon's bound, showing a bound of $O(\log^2 n)$. We prove an upper bound of $(\log n)^{1+o(1)}$ for this maximum possible average degree when there is no rainbow cycle. Our result is tight up to the $o(1)$ term, and so it essentially resolves this question. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non-abelian groups