11 problems

Let $F$ be a graph on $[n]$, let $e=e(F)$ be its number of edges, and let $p_c(F)$ denote the threshold probability for containing a copy of $F$ in a random graph. Kahn–Narayanan–P…

Let $G_{n,p=c/n}$ be the Erdős–Rényi random graph, and let $c_k$ denote the threshold for the appearance of a non-empty $k$-core, meaning a maximal subgraph with minimum degree at…

Let $k$ be an odd integer, and let $H$ be an $n$-vertex $k$-uniform hypergraph containing no $2$-regular subgraphs. A full $k$-star consists of all $k$-edges containing a fixed cen…

Let $d\geq 3$ and let $F=F(n)$ be a sequence of $d$-regular graphs on $[n]$, with $n\in\mathbb{N}$. For a subgraph $\tilde F\subset F$, let $\partial_e(\tilde F)$ denote its edge b…

Let $k\geq 2$ be an integer, let $\lambda\geq 1$, and let a $k$-uniform linear hypergraph be a hypergraph in which every edge has $k$ vertices and any two distinct edges intersect…

Let $d(r,n)$ be the smallest average degree such that every $n$-vertex graph with average degree at least $d(r,n)$ contains an $r$-regular subgraph. Quadratic-logarithmic conjectur…

An $r$-regular graph is a graph in which every vertex has degree $r$; the average degree is the average of the vertex degrees. Chvátal's conjecture. Some average degree condition d…

Let $G$ be an $n$-vertex graph with average degree $d$, and let $r$ be a positive integer. The authors' conjecture. For some constant $C=C(r)$, if … then $G$ contains an $r$-regula…

Let $s\geq 3$ be an integer and let $\varepsilon>0$. Bounded-size regular-subgraph conjecture. There is a positive integer $t$ such that, for all sufficiently large $n$, every $n$-…

Let $s\geq 3$ be an integer. For sufficiently large $n$, let $G$ be an $n$-vertex graph with average degree at least $d$, where $C\log\log n\leq d\leq n^{\frac{s-2}{s}}$. Regular-s…

Let $k\geq 3$ be an integer. An $n$-vertex $k$-uniform hypergraph is said to have no $2$-regular subgraphs on $2k$ vertices if it contains no subhypergraph on exactly $2k$ vertices…