4 problems

Győri's conjecture. For every fixed $r\geq 3$,

Let $G(n,p)$ be the Erdős–Rényi random graph, let $p\in(0,1)$ and $C\in\mathbb{N}_{\geq 3}$ be constants, and let $k_0$ be the least integer such that the expected number of $k_0$-…

Let $r\geq t\geq 3$ and $k\geq 2$, and let $K_{t}$ denote the complete graph on $t$ vertices. For positive integers $n_1,\ldots,n_r$, write $K_{n_1,\ldots,n_r}$ for the complete…

Let $n,r\in\mathbb{N}$ with $r$ dividing $n$, and let $G$ be a graph on $n$ vertices whose degree sequence is $d_1\leq\cdots\leq d_n$. Balogh–Kostochka–Treglown conjecture. If … fo…