22 problems

Erdős–Simonovits–Sós conjecture.

Let $H$ be a $k$-uniform hypergraph, let $M_s$ denote a matching with $s$ edges, let ${\rm ar}(n,k,M_s)$ be the anti-Ramsey number of $M_s$ in the complete $k$-uniform hypergraph o…

Let $k\geq 3$ be an integer, and let $\chi_S(n,e,L)$ denote the minimum number of colors in an edge-coloring of an $n$-vertex graph with at least $e$ edges in which every copy of…

Jahanbekam–West conjecture. One has

Mubayi's conjecture. The Turán density of $F_4$ is

Let $k,t,n$ be positive integers, let $G=kP_4\cup tP_2$, and let $S\subseteq E(G)$ consist of internal edges of the $P_4$ components. A set of edges is neutral for the anti-Ramsey…

Rainbow anti-Ramsey conjecture. The rainbow anti-Ramsey number for $(t+2)$ vertex-disjoint triangles satisfies

Let $n$ and $k$ be integers with $n\geq 3k$, let $kC_3$ be the vertex-disjoint union of $k$ triangles, and let $G_1(n,k),\ldots,G_4(n,k)$ be the four explicitly defined edge-colore…

For positive integers $n$ and $k$ with $n\geq 3k$, let $kC_3$ denote the vertex-disjoint union of $k$ triangles, and let $ar(n,kC_3)$ be the maximum number of colors in an edge-col…

For positive integers $n$ and $k$, let $ar(n,C_k)$ be the maximum number of colors in an edge-coloring of the complete graph $K_n$ containing no rainbow cycle $C_k$. Erdős–Sós–Simo…

Let $r\geq 3$, and let $M_k$ be a matching consisting of $k$ pairwise disjoint $r$-edges. Write ${\rm ar}(n,r,M_k)$ for the anti-Ramsey number and ${\rm ex}(n,r,M_{k-1})$ for the T…

Özkahya and Young's conjecture.

Let $P_n^t$ be the hypergraph with vertex set $[1,n]$ and edge set consisting of the $t$-term arithmetic progressions. For a hypergraph $H$, let $\overline{\chi}_b(H)$ denote the l…

Let $s\ge 3$. A graph is $s$-rainbow common if, among all $s$-colorings of the edges of $K_n$, the maximum number of rainbow copies of the graph is asymptotically achieved by color…

Let $K_{n_1,n_2,\ldots,n_r}$ be a complete $r$-partite graph with $r\ge 3$, $n_1\ge n_2\ge\cdots\ge n_r$, and $n_1,\ldots,n_r$ sufficiently large relative to $k$. Let…

Let $T_k$ be a tree with $k$ edges, and let $pr(K_n,T_k)$ denote the maximum number of colors in an edge-coloring of $K_n$ containing no properly colored copy of $T_k$. Properly co…

Let $T$ be a tree with $k$ edges, and let $ar(K_n,T)$ denote the maximum number of colors in an edge-coloring of $K_n$ containing no rainbow copy of $T$. Jiang and West's conjectur…

A spider is a tree with at most one vertex of degree greater than $2$; its legs are the paths from its center to its leaves. Let $T$ be a spider with $p\ge 3$ legs, each of length…

Let $T_k$ be a tree with $k$ edges, and let $ar(n,T_k)$ denote the anti-Ramsey number for $T_k$. Jiang–West's conjecture. … The conjecture is presented as a proposed improvement to…

Let $G$ be a connected graph on $n_{0}\ge 3$ vertices, and let $t\ge 1$. The anti-Ramsey number $ar(K_n,tG)$ is defined as the maximum number of colors in an edge-coloring of $K_n$…

Let $K_{2n}$ be a complete graph with a proper edge-coloring using exactly $2n-1$ colors. A multicolored $2$-factor is a $2$-factor whose edges have pairwise distinct colors. The r…

Let $P_k$ be the path on $k$ edges, and let $AR_{loc}(P_k)$ denote its local anti-Ramsey number. Gyárfás et al.'s conjecture. The number $AR_{loc}(P_k)$ should be equal to $k+c$ fo…