134 problems

For an integer $r\geq 5$, let $D_2^\#(K_r)$ denote the maximum number of edges that must be removed to make an $n$-vertex $K_r$-free graph bipartite. Sudakov's conjecture. … The fo…

Gerbner–Palmer conjecture. For every graph $H$, there exists $k_0=k_0(H)$ such that $H$ is $K_k$-Turán-good for every $k\geq k_0$.

Let $r\ge 2$. An $r$-graph is a hypergraph whose edges have size $r$. Let $\mathbb{C}_{r}$ be the collection of $r$-graphs consisting of three edges $A,B,C$ such that … Let…

The $P_4^r$ upper-bound conjecture.

Let $\mathcal{F}$ be a finite family of graphs, let $\chi(\mathcal{F})$ denote its chromatic number, and let $\operatorname{ex}(n,\mathcal{F})$ be the maximum number of edges in an…

Let $K_{s,t}^{(3)}$ denote the complete $(s,t)$-partite $3$-uniform hypergraph, and let $\mathrm{ex}(n,K_{s,t}^{(3)})$ be the maximum number of edges in an $n$-vertex…

Let $t\geq 2$ be an integer, and let $H$ be a $K_{2,t}$-free bipartite graph such that every vertex in one of the parts of $H$ has degree at most $t$. Conlon–Janzer–Lee conjecture.…

Tyomkyn–Uzzell's conjecture. For $k\geq 3$ and $n\geq k+1$, except when $k=3$ and $n=8$, every $n$-vertex graph $G$ such that $G_k$ is triangle-free satisfies

Let $C_k$ denote the cycle on $k$ vertices, and let $\operatorname{ex}_{\mathcal{P}}(n,C_k)$ be the maximum number of edges in an $n$-vertex planar graph containing no copy of…

Let $G$ be an $n \times n \times n$ tripartite graph, and let $\delta(G)$ denote its minimum degree. For an integer $t\geqslant 2$, let $\tau$ be the additive term in the condition…

Let $f_r(n,t)$ be the smallest number such that every $K_{r+1}$-free graph $G$ with at least $e(T_r(n))-t$ edges can be made $r$-partite by deleting $f_r(n,t)$ edges. A pentagonal…

Let $r\geq 3$, $k\geq h$, and let $H$ be an $r$-color-critical graph with $h$ edges. For sufficiently large $n$, an $n$-vertex $k$-color extremal multigraph of $H$ is either formed…

Let $C_{13}$ denote the crown on 13 vertices, and let $ex^{lin}_3(n,C_{13})$ be the maximum number of edges in a linear 3-uniform hypergraph on $n$ vertices that contains no copy o…

Day–Falgas-Ravry–Treglown conjecture. For all integers $a,r,s,d$ with $a,r\geq 1$, $d\in[0,a-1]$, $s\geq(r-1)(d+1)+2$, and all sufficiently large $n$,

Let $C_k$ denote the cycle of length $k$, and let $ex_{\mathcal{P}}(n,C_k)$ be the maximum number of edges in an $n$-vertex planar graph containing no copy of $C_k$. Cranston–Lidic…

Let $H$ be a graph, and let $f(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of $H$ in a $2$-edge-coloring of the complete graph $K_n$. Keevash–S…

Let $F=kP_{\ell}$ be a linear forest consisting of $k\geq 2$ vertex-disjoint paths of length $\ell$, with $\ell\neq 3$. Let $G_F(n)$, $\delta_F$, and $\mu$ be as in Theorem BK, whe…

Let $R$ and $B$ be graphs on the same vertex set of size $n$, and let $H$ be a red-blue coloring of the edges of the complete graph $K_{k+1}$. Define $\mathrm{mex}(n,H)$ to be the…

Let $n$ be a positive integer. A cancellative $3$-graph is a $3$-uniform hypergraph in which $A\cup B=A\cup C$ for edges $A,B,C$ implies $B=C$. For a partition…

Even-cycle degree-power Turán conjecture. For every $p>1$,

Tree Turán number conjecture.

For a graph $G$, let $N(C_q,G)$ denote the number of copies of the cycle $C_q$ in $G$, and let $\operatorname{ex}(n,C_q,\mathcal C_{\ge L+1})$ be the maximum possible value of…

Zhou and Li's conjecture. The same equality should hold whenever $n\geq 3k$; moreover, the extremal digraphs should be precisely the transitive Turán digraphs…

Let $F$ be an edge-color-critical graph with chromatic number $\chi(F)=r+1\ge 3$. For a positive integer $n$, let $\mathrm{EX}_{r+1,\rho}(n,F)$ be the family of non-$r$-partite $F$…

Let $H$ be a graph with chromatic number $\chi(H)\ge 3$, and let $P_k$ denote the path on $k$ vertices. For a family of graphs $\mathcal{F}$, write $\mathrm{ex}(n,\mathcal{F})$ for…