6 problems

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…