15 problems

Fix $k\ge 2$. Let $\delta>\delta_0>0$ and let $n$ be sufficiently large. For a graph $G$, write $D_2(G)$ for the maximum number of edges in a bipartite subgraph of $G$, and call a…

Let $k$ be a positive integer, and let $C_{2k+1}$ denote the odd cycle of length $2k+1$. A blowup of $C_{2k+1}$ is obtained by replacing its vertices with independent sets and its…

Let $k$ and $\ell$ be positive integers with $\ell>k$. Forbidding $C_{2k-1}$, or more generally a graph $F$ with $\gamma(F)=k$, consider the problem of maximizing the number of cop…

Fix $k\geq 2$, and let $\delta$ be small enough. For any $\delta>\delta_0>0$ and sufficiently large $n$, consider every $C_{2k-1}$-free graph $G$ on $n$ vertices satisfying … Korán…

Let $K_{k,k,k}$ denote the complete tripartite graph with three parts of size $k$, and let $r(F;q)$ be the least $n$ such that every $q$-edge-coloring of the complete graph on $n$…

Cycle obstruction conjecture. There exists an integer $t$ such that, for every $s$, there is a graph $G$ satisfying

Bowen–Hansberg–Montejano–Müyesser's conjecture. For every $r,t\in\mathbb{N}$, any $r$-edge-colouring of a sufficiently large graph on $n$ vertices with $\Omega(n^{2-1/t})$ edges in…

Fox–Luo–Wigderson's conjecture. There exists a graph $H$ and integers $r,t\geq 2$ such that there are graphs $G_1,G_2,\dots$ with $G_i\stackrel{r}{\to}H$ for every $i$ and

Let $\mathcal{T}\in\mathcal{T}_{s,t}(a,b)$, where $b\leq a<r-1$, and set $\sigma=\sigma(\mathcal{T})=\min\{s,t\}$. Let $H$ be a $\mathcal{T}$-free $n$-vertex $r$-graph. General tre…

Path-blowup conjecture. The family $\Psi_{k-1}(n,r)$ gives the correct asymptotic of the Turán number in all the above cases.

Let $G \overset{r}{\longrightarrow} H$ mean that every $r$-edge-coloring of $G$ contains a monochromatic copy of $H$, and let $\mathrm{B}(G \overset{r}{\longrightarrow} H;t)$ be th…

Let $G$ and $H$ be graphs and let $r$ be an integer such that $G \overset{r}{\longrightarrow} H$. For an integer $t$, let $\mathrm{B}(G \overset{r}{\longrightarrow} H;t)$ be the mi…

Let $G$ and $H$ be graphs, let $r\geq 2$, and write $G[n]$ for the $n$-blowup of $G$. Write $G\stackrel{r}{\rightarrow}H$ when every $r$-edge-colouring of $G$ contains a monochroma…

Logarithmic balanced-clique conjecture. There is an absolute constant $c>0$ such that $G$ contains a copy of $K_{t,t,t}$ with

Skew blow-up conjecture. There is a constant $c>0$ such that