17 problems

Jungić–Licht–Mahdian–Nešetřil–Radoičić conjecture. The quantity $T_k$ satisfies

For an integer $r\geq 1$ and a subset $A\subseteq [n]$, let $c\colon A\to [r]$ be an $r$-coloring. A solution $(x_1,x_2,y_1,y_2)$ to $x_1+x_2=y_1+y_2$ in $A$ is rainbow if the four…

Let $[k]^n$ be the $k$-ary hypercube. A geometric line is a line in the hypercube in the geometric sense, and the anti-Hales–Jewett number is the minimum number of colors guarantee…

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…

Rainbow girth conjecture. The rainbow girth is at most

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

Jungić et al.'s conjecture. The lower bound for $T_k$ is correct:

Let $\vee_2$ be the poset consisting of one element below two incomparable elements, and let $\wedge_2$ be its dual. Let $f(n,k,P)$ be the maximum possible size of the smallest col…

Let $P_3$ be a chain of three elements, and let $f(n,k,P)$ be the maximum possible size of the smallest color class in a $k$-coloring of the Boolean lattice $B_n=2^{[n]}$ that avoi…

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…

Let $P$ be a poset and let $F(n,l,P)$ denote the maximum size of a color class in an $l$-coloring of the Boolean lattice $B_n=2^{[n]}$ that avoids a rainbow copy of $P$. Suppose th…

Let $T_k$ be the minimal number $t\in\mathbb{N}$ such that every equinumerous $t$-coloring of $[tn]$, for every $n\in\mathbb{N}$, contains a rainbow arithmetic progression of lengt…

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

Anti-van der Waerden drop conjecture. For positive integers $n$ and $k$,

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

Let $n$ be an integer that is not a power of $2$. A rainbow-free $3$-coloring of the cyclic group $\mathbb{Z}/n\mathbb{Z}$ is a coloring with three nonempty color classes containin…