7 problems

Converse of Sun's characterization. $H$ is rainbow-common if and only if $H$ is a forest.

Let $r\ge 2$ be an integer and let $n$ be sufficiently large. An $r$-coloring of a subset of $[n]$ is rainbow $[1,-1,1,-1]$-free if it contains no rainbow $4$-element set satisfyin…

Even-clique threshold conjecture. For every real $d>0$ and every integer $r\geq 5$, the threshold for the property

Let $A_k$ be an antichain of size $k$, and let $f(n,c,A_k)$ be the maximum size of a color class in a $c$-coloring of $B_n=2^{[n]}$ avoiding a rainbow copy of $A_k$. Sharp exponent…

Let $A_k$ be an antichain of size $k$, and let $\bigvee_s$ and $\bigwedge_s$ denote the posets consisting respectively of a common minimal or maximal element together with $s$ inco…

Power-of-three anti-van der Waerden conjecture. For every nonnegative integer $m$,

Logarithmic upper-bound conjecture. There exists a constant $C$ such that