20 problems

For $\delta>0$, let $\rho(3,7,\delta)$ be the limiting maximum edge density of an $n$-vertex graph with independence number at most $\delta n$ that admits a red/blue coloring with…

For $\delta>0$, let $\rho(3,6,\delta)$ be the limiting maximum edge density of an $n$-vertex graph with independence number at most $\delta n$ that admits a red/blue coloring with…

Let $r(3,q)$ denote the two-color Ramsey number for a triangle and a clique of order $q$. For $q\geq 2$, the Ramsey–Turán density $\rho(3,2q-1)$ is defined using graphs admitting a…

Let $\mathcal{L}: c_1x_1 + \ldots +c_k x_k = 0$ be a homogeneous linear equation with $k \ge 3$, where $c_1,\ldots,c_k \in \mathbb{Z} \setminus \{0\}$ satisfy … but there exists a…

For each integer $q\ge 5$, let $\rho(3,m)$ denote the Ramsey–Turán density for graphs admitting a 2-edge-coloring with no monochromatic blue $K_3$ and no monochromatic red $K_m$, a…

Let $\rho(3,6,\delta)$ denote the asymptotic maximum edge density of an $n$-vertex graph with independence number at most $\delta n$ admitting a 2-edge-coloring with no monochromat…

Let $p>q\geq 3$ be integers, and let $\mathcal{G}(n;s,t)$ denote the class of constructions with parameters $s$ and $t$ used for the generalized Ramsey–Turán problem. Balogh–et al.…

Fix integers $s$ and $t$ with $3\leq s\leq t-2$. A weighted graph admitting a $(b,t-1-b)$-partition has $b$ vertices and $t-1-b$ parts satisfying the defining partition conditions…

Let $s$ and $t$ be integers with $3\leq s\leq t-2$. An extremal graph for the generalized Ramsey–Turán density $\varrho_s(K_t)$ is a graph admitting either an $(s,t-1-s)$-partition…

Connecting-barrier conjecture. Given $\mu>0$ and $r\geq 4$, there exists $\alpha>0$ such that the following holds for sufficiently large $n$. If

Staden–Treglown conjecture. For every $\mu>0$, there exist $\alpha>0$ and $0in \mathbb{N}$ such that the following holds. For every $n$-vertex graph $G$ with $\geq n0$, if

Let $G$ be an asymptotically extremal graph for the Ramsey–Turán density $\varrho_p(q)$, where the $p$-independence number is the largest size of a vertex set inducing a $K_p$-free…

Let $r$ and $k$ be integers satisfying $1\le r\le k$ and $k\ge 3$. The Ramsey–Turán tiling function ${\bf{RTT}}_r(K_k)$ is defined as the asymptotic minimum-degree threshold for fo…

Erdős–Hajnal–Simonovits–Sós–Szemerédi conjecture. The asymptotic extremal graphs $G$ for $\varrho_p(q)$ have a partition

Let $n$ and $s$ be integers with $n/3<s\le n/2$, and let $\mathrm{ex}(n,s)$ denote the maximum number of edges in a triangle-free graph on $n$ vertices with independence number at…

Reiher–LPR conjecture. One has

Ramseye1Ture1n density formula. The function $f$ is given by

Let $n$ and $s$ be integers with $n s\e 0$. For $k\e 2$, define … Define $g(n,s)$ by … Here $\operatorname{ex}(n,s)$ denotes the maximum number of edges in a triangle-free $n$-vert…

Let $R(3,s)$ be the smallest integer $N$ such that every red-blue colouring of $K_N$ contains a red $K_3$ or a blue $K_s$. Let $\varrho(K_3,K_{2s})$ denote the multicolour Ramsey–T…

Let $t>s\ge 3$. For $t\le 2s-1$, let $\mathcal{H}(n,s,t)$ be the family of graphs from the stated construction: partition the vertex set into $V_1\cup\cdots\cup V_s$, use an extrem…