75 problems

Let $G$ be an $n$-vertex $C_{\ell}$-free plane graph, meaning that $G$ contains no cycle of length $\ell$, where $\ell\geq 7$. Write $e(G)$ for the number of edges of $G$. Asymptot…

The $P_4^r$ upper-bound 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…

Let $\mathcal{H}$ be an $r$-graph, and let $d_{\mathcal{H}}(e)$ denote the codegree of an $(r-1)$-set $e$. Define … For an $r$-graph $\mathcal{H}$, let … Given a balanced partition…

Let $G$ be a graph on $n$ vertices, let $T_r(n)$ denote the Turán graph, let $N(v)$ denote the neighborhood of a vertex $v$, let $d(v)$ denote its degree, and let $bb(G)$ denote th…

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 $TP_4$ be the triangular pyramid with $4$ layers, and let $\textup{ex}(n,TP_4)$ denote the maximum number of edges in an $n$-vertex graph containing no copy of $TP_4$. Ghosh et…

Győri et al.'s linear Turán conjecture. If $G$ contains no linear path of length $k\ge 5$, then the number of edges in $G$ is at most

Let $P_4^r$ be the $r$-uniform linear path with four edges, and let $ex_r^{\mathrm{lin}}(n,P_4^r)$ denote the maximum number of edges in an $n$-vertex linear $r$-uniform hypergraph…

The matching-restricted Berge-clique conjecture.

For integers $r$ and $s\geq r-1$, let $G_1,\ldots,G_s\subseteq\binom{[n]}{2}$ be graphs whose edges are $(r-1)$-star edge-colored and whose underlying graphs are $K_r$-free. Here…

Let $F$ be a $3$-graph, let $\chi(F)$ denote its chromatic number, and let $q(F)$ be the parameter defined by the ordered link-graph chromatic numbers in the source. For…

The spectral stability conjecture. There exists $n_{0}$ such that for any $\mathcal{F}$-free $r$-graph $H$ on $n\geq n_{0}$ vertices,

Updated Turánability conjecture. The oriented graph $H$ is Turánable if and only if there are integers $r$ and $k$ such that

DeBiasio–Han–Lo–Molla–Piga–Treglown conjecture. The oriented graph $H$ is Turánable if and only if there is an $s\in\mathbb{N}$ such that $H\subset D_s$.

Let $g_k(n)$ be the maximum number of edges in an $n$-vertex iterated blowup of a $k$-uniform edge. A $k$-partite coloring of the complete graph $K_n$ is a coloring arising from a…

Let $g_k(n)$ be the maximum number of edges in an $n$-vertex iterated blowup of an edge. Thus $g_k(n)=0$ for $n<k$, and for $n\geq k$, … where the maximum is over compositions of…

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 $k\ge 2$, and let $F$ be a color-$k$-critical graph with chromatic number $\chi(F)=r+1\ge4$. Define $H_{n,r,k}:=K_{k-1}\vee T_{n-k+1,r}$, where $T_{n-k+1,r}$ is the $r$-partite…

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

Katona–Xiao conjecture. If $k$ is odd and $(k-1)\mid n$, then the disjoint union of $n/(k-1)$ copies of $T(k-1,m-1)$ gives the maximum number of edges in a graph containing neither…

Let $r$ and $k$ be integers satisfying … For an $r$-uniform hypergraph on $n$ vertices, let $\operatorname{ex}^{\mathrm{conn}}_r(n,\mathcal{B}P_k)$ denote the maximum number of edg…

Let $G$ be a graph with $m$ edges. A graph is $K_{r+1}$-free if it contains no complete subgraph on $r+1$ vertices. Let $q(G)$ denote the largest eigenvalue of the signless Laplaci…

Let $H$ be a graph with chromatic number $\chi(H)\le k$ and let $k\ge 3$. Write $T(n,k)$ for the $n$-vertex Turán graph with $k$ parts, and let $\mathcal N(H,G)$ denote the number…

Let $F$ be a graph. Write $\operatorname{EX}(n,F)$ for the family of $F$-free graphs attaining the extremal number, and let $\operatorname{SPEX}(n,F)$ denote the family of $F$-free…