4 problems

Let $K_{\ell+1}^{(r)+}$ be the $r$-uniform expansion of the complete graph on $\ell+1$ vertices, let $M_{s+1}^{(r)+}$ be an $r$-uniform matching with $s+1$ edges, and let…

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 $K^{(r)+}_{\ell+1}$ be the $r$-uniform expansion of the complete graph on $\ell+1$ vertices, let $M^{(r)+}_{s+1}$ be an $r$-uniform matching with $s+1$ edges, and let…

Let $G$ be a graph with chromatic number $\chi(G)=\ell>r$, and let $U$ be an independent set of $G$ such that deleting $U$ results in a graph $G_0$ of chromatic number $\ell-1$. Su…