8 problems

Simion's non-crossing matching conjecture. For every positive integer $n$,

The q-analogue conjecture. For every such $n$ and $k$,

Homogeneous necklace matching conjecture.

Let $L\in\mathcal{N}(N)$ be a balanced necklace with $N=2n$ beads, and let $\operatorname{mono}(L)$ denote the number of maximal monochromatic arcs. In the heterogeneous model, a p…

Kynčl–Pach–Tóth refined conjecture. For every constant $k$, there is always a proper matching of size at least

Heterogeneous necklace matching conjecture. In the heterogeneous model, every balanced necklace has a proper matching of size at least $2n/3-o(n)$; equivalently, $\mu=2/3$. This co…

The rationality phenomenon for finite submatching events. For $n\ge k$, there is an even polynomial $Q_{\pi_0}(n)$ of degree $k(k+1)$ with dyadic rational coefficients such that

The rationality phenomenon for submatching events. For any $\pi_0\in\textnormal{NC}_{n}$, the probability of the associated submatching event is a dyadic rational number that is co…