11 problems

Perhaps this is really a Turán type problem and not a Ramsey problem. In other words, if $m$ is sufficiently large and $1 < a_1 < a_2 < \cdots < a_\ell \leq m$ is a sequence of int…

An old problem of R.L. Graham and myself states: Is it true that if $m_k$ is sufficiently large and we colour the integers $2 \leq t \leq m_k$ by $k$ colours then … is always solva…

One last Ramsey type problem: Let $n_k$ be the smallest integer (if it exists) for which if we colour the proper divisors of $n_k$ by $k$ colours then $n_k$ will be a monochromatic…

Polynomial-generation conjecture. For every positive integer $\ell$, there exist $x,y,z\in\mathbb{N}^*$ such that

Let $k$ be a positive integer and let $p$ be an odd prime such that $k+1=p^2$. Let $f(k,-1)$ denote the least board size for which Maker can force a solution to the unit-fraction e…

Let $f^*(k,-1)$ be the least board size for which Maker can force a solution with distinct variables in the Maker–Breaker game for the equation $1/x_1+\cdots+1/x_k=1/y$, and let…

Erdős–Graham conjecture. There is a unit-fraction representation $S$ that is monochromatic, apart from the trivial representation $S=\{1\}$.

Schinzel's generalized Erdős–Straus conjecture. Given $m\geq 3$, for all $n>n_0(m)$ there exist $u_i\in\mathbb{N}$ satisfying the equation above.

Exceptional-set conjecture. For some such quadratic integer rings, the equation has a decomposition for every $n$ outside an exceptional set. This is a restatement of the paper's e…

Bradford–Ionascu conjecture. Such a decomposition exists with $a,b,c\in\mathbb{Z}[i]$ such that the real and imaginary parts of each of $a$, $b$, and $c$ are either both nonnegativ…

Let $r\geqslant 1$ be an integer, and let $X$ be a positive parameter. Spacing of unit fractions. The number of $x_1,\dots,x_{2r}\in[X,2X)$ satisfying … is…