7 problems

Monotonicity conjecture for Issai numbers.

Abbott–Wang conjecture. For every $r\in\mathbb{N}^*$, one has

Let $S(3;s,t,u)$ denote the least positive integer $N$ such that every coloring of $[1,N]$ with three colors contains, for some color corresponding to $s$, $t$, or $u$, a monochrom…

For $n\ge 2$, let $S(n)$ denote the $n$th Schur number. The recursive Schur-number upper-bound conjecture. … This conjecture follows from the preceding conjecture on $L(n)$ togethe…

For $n\ge 2$, let $S(n)$ be the $n$th Schur number. Let $L(n)$ be the least length such that every sequence $A$ in $\mathbb N_+$ with $|A|=L(n)$ and average $\mu(A)\le n$ satisfies…

A Schur triple in $[1,n]$ is a monochromatic triple $(x,y,z)$ satisfying $x+y=z$ under an $r$-coloring of $[1,n]$. Let the minimum be taken over all $r$-colorings of $[1,n]$, with…

For an $r$-coloring of $[1,n]$, a Schur number $s(r)$ is the maximal integer $n$ such that there exists an $r$-coloring of $[1,n-1]$ with no monochromatic solution to $x+y=z$. The…