11 problems

For integers $k,r\geq 1$, let $w(k;r)$ denote the least positive integer $N$ such that every coloring of $\{1,\ldots,N\}$ with $r$ colors contains a monochromatic arithmetic progre…

For integer $k\ge 3$, let $w(3,k)$ be the smallest $N$ such that every blue-red coloring of $[N]:=\{1,\dots,N\}$ contains either a blue arithmetic progression of length $3$ or a re…

The van der Waerden number $W(r,k)$ is the smallest integer $n$ such that every red and blue coloring of $[n]:=\{1,2,\dots,n\}$ contains either a monochromatic red $r$-term arithme…

Let $k\geqslant 3$ be a positive integer, and let $w(3,k)$ denote the least $N$ such that every red-blue colouring of $[N]=\{1,\ldots,N\}$ contains either a blue 3-term arithmetic…

Let $W(r,k)$ denote the van der Waerden number. Quadratic growth-ratio conjecture. As $k$ grows large with $k>r$, … This is the simplified form obtained in the source under the add…

Let $W(r,k)$ denote the van der Waerden number, and write $W(r,k) = (r)^{n_r}$ and $W(r+1,k)=(r+1)^{n_{r+1}}$ for the corresponding exponents. Growth-ratio conjecture. As $r$ grows…

Let $w(2;3,t)$ denote the van der Waerden number for 2-colourings with monochromatic progressions of lengths $3$ and $t$. Quadratic upper-bound conjecture. For $t\geqslant 3$, … Th…

For $t\geqslant 3$, let $S(t)$ be the set of good partitions considered in the paper, and for $B\in S(t)$ let $n_1(B)$ denote the number of $1$'s in $B$. One-count bound conjecture…

For $t\geqslant 3$, let $S(t)$ be the set of good partitions considered in the paper, and for $B\in S(t)$ let $n_{00}(B)$ denote the number of occurrences of $00$ in $B$. Double-ze…

Let $w(2;3,t)$ denote the least integer $N$ such that every 2-colouring of $\{1,\ldots,N\}$ contains a monochromatic arithmetic progression of length $3$ in one colour or length…

For $s\geq 2$, let $G(3,s)$, $w(3,s)$, $M(3,s)$, $w_1(3,s)$, $w^*(3,s)$, and $w(3;s)$ be the functions defined in the paper that measure the relevant van der Waerden-type quantitie…