4 problems

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…