29 problems

Andrásfai's conjecture. For all integers $t$ and $n$, the extremal Ramsey–Turán graph is a canonical blow-up of an Andrásfai graph; specifically,

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.…

Consider graphs in $\mathcal{G}(n;s,t)$ that asymptotically achieve the maximum number of copies of $K_q$ in an $n$-vertex $K_p$-free graph under the relevant Ramsey–Turán constrai…

Let $\mathfrak{R}_q(p)$ denote the generalized Ramsey–Turán density for copies of $K_q$ in a $K_p$-free graph. The results for small $q$ suggest that, for sufficiently large $p$, t…

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…

Andr'asfai's conjecture. For all integers $k$, $s$, and $n$ such that $s>n/3$ (equivalently, in the formulation above, whenever $\frac{kn}{3k-1}\le s\le\frac{(k-1)n}{3k-4}$),

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…

For a graph $H$ and a function $f(n)$, let $\rho\tau(H,f(n))$ denote the limiting maximum edge density of an $H$-free graph on $n$ vertices whose independence number is at most…

For a graph $H$ and a function $f(n)$, let $\rho\tau(H,f(n))$ denote the limiting maximum edge density of an $H$-free graph on $n$ vertices whose independence number is at most…

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

Let $n$ and $s$ be positive integers with $nge s$, and let $\mathrm{ex}(n,s)$ be the maximum number of edges in a triangle-free graph $G$ on $n$ vertices with $\alpha(G)\le s$. For…

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 $R(3,s)$ 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 $\varrho(K_3,K_{2s-1})$ denote the multicolour Ramsey…

The $K_6$ equality conjecture. Equality should hold: