4 problems

Let $C_{2k}$ be the cycle of length $2k$, and let ${\operatorname{ex}^\star}(n,C_{2k})$ denote the maximum number of edges in an edge-coloured graph on $n$ vertices with no properl…

Let ${\mathrm{ex}}^{\sum}_r(n,k,K_\ell^{(r)+})$ denote the rainbow hyper-Turán number for the $r$-uniform expansion $K_\ell^{(r)+}$, and let $t_r(n,\ell-1)$ be the number of edges…

Let $P_k$ denote the path on $k$ vertices, and let $\operatorname{ex}_{\mathcal{P}}^*(n,P_k)$ be the maximum number of edges in an $n$-vertex planar graph that has a proper edge-co…

For an integer $k$, let $P_k$ denote a path of length $k$, with length defined as its number of edges. Let $ex^{*}(n,F)$ be the maximum number of edges in a properly edge-coloured…