4 problems

Grzesik–Janzer–Nagy conjecture for even cycles. For every pair of integers $r,k\geq2$,

For an integer $r\geq 2$ and a graph $H$, let $f_r(n,H)$ be the smallest number $C$ such that some proper edge-colouring of $K_n$ with $C$ colours contains no $r$ vertex-disjoint c…

Let $C_{2k}$ denote the cycle of length $2k$, and let $\mathrm{ex}^*(n,H)$ be the maximum number of edges in a properly edge-coloured $n$-vertex graph containing no rainbow copy of…

Let $\ell\geqslant2$, and write … for the maximum number of edges in an $n$-vertex graph containing none of the cycles $C_3,C_4,\ldots,C_{2\ell}$. Erd s–Simonovits conjecture. … Th…