22 problems

Erdős's conjecture.

Rödl–Szemerédi conjecture. For every $d\geq 3$ there exist $\varepsilon>0$ and a sequence of graphs $(G_n)$ on $n$ vertices and maximum degree at most $d$ such that

Given a graph $H$ and a function $c3:E(H)\to\mathbb{N}$, the subdivision $H^{\sigma}$ is obtained by replacing each edge $e\in E(H)$ with a path of length $\sigma(e)$. For a graph…

Let $K_t$ be the clique on $t$ vertices. The online-to-size Ramsey conjecture. … Here $\tilde{r}(G,H)$ is the online size Ramsey number and $\hat{r}(G,H)$ is the size Ramsey number…

Let $cDelta\textgreater=1$, and let $T$ be a tree on $n$ vertices with maximum degree $cDelta(T)\textgreater= cDelta$. For $q\textgreater=1$, write $crhat_{\mathrm{ind}}^q(T)$ for…

Vito–Silaban's conjecture. For $n\ge 5$ and $m\ge 3$,

Baskoro–et al.'s conjecture. The upper bound

Let $nK_{1,p}$ be the disjoint union of $n$ copies of the star $K_{1,p}$, and let $K_{1,m}$ be the star with $m$ edges. The connected size Ramsey number $\hat r_c(G_1,G_2)$ is the…

Let $k$ be a positive integer. Exponential gadget conjecture. There is a graph $G$ with $e^{O(k)}$ edges such that every $k$-coloring of its edges contains a monochromatic odd cycl…

Let $C_n$ be the cycle on $n$ vertices, and let $\hat{r}^k_{\operatorname{ind}}(C_n)$ denote the smallest number of edges in a graph whose every $k$-coloring contains a monochromat…

Let $H$ be an $n$-vertex graph, and let $\hat{r}^2_{\operatorname{ind}}(H)$ denote the smallest number of edges in a graph whose every $2$-coloring contains a monochromatic copy of…

Even-cycle online Ramsey conjecture.

Let $H$ be an $n$-vertex graph of maximum degree $3$, and let $\hat{r}(H)$ denote the minimum number of edges in a graph that is Ramsey for $H$. Rödl–Szemerédi conjecture. There ex…

Conjecture on the triangle–path online size Ramsey number.

Let $B_n^{(k)}$ denote the book graph consisting of $k$ triangles sharing a common edge, with the shared edge contained in $k$ pages, each having $n$ vertices in the relevant book…

Let $K_{s,t}$ denote the complete bipartite graph with parts of sizes $s$ and $t$, and let $hat r(H)$ denote the size Ramsey number of a graph $H$. For functions of the parameters,…

Multicolor star-forest conjecture. The multicolor size Ramsey number satisfies

Dudek–La Fleur–Mubayi–Rödl's conjecture. The size-Ramsey number of tight paths is linear in the number of vertices $n$, that is, for each fixed $k$, there is a constant $C_k$ such…

Let $n\ge 5$ and $m\ge 3$. For graphs $G$ and $H$, write $\hat{r}(G,H)$ for their size Ramsey number, and let $K_2$ be a single edge and $P_m$ a path on $m$ vertices. The graph…

For positive integers $D,q,r,n$, let $USR(D,q,r,n)$ be the smallest number of edges of a graph $G$ such that … for every graph $H$ on $n$ vertices with maximum degree at most $D$.…

Let $K_{1,k}$ be the star with $k$ edges, and let $K_n$ be the complete graph on $n$ vertices. The size Ramsey number $\hat{r}(K_{1,k},K_n)$ is the minimum number of edges in a gra…

Rödl–Szemerédi conjecture. There is an $\varepsilon>0$ such that