The non-nested and non-separated matching conjectures for ordered complete graphs

Let $t,n$ be positive integers and set

$$m=(t-1)(n-1)+2n.$$

Consider a $t$-coloring of the edges of the ordered complete graph $K_m$, whose vertices are linearly ordered. A matching is a set of pairwise vertex-disjoint edges; it is non-nested if no two edges are nested and non-separated if no two edges are separated in the vertex order. Ordered matching conjecture. Every such coloring contains (i) a monochromatic non-nested matching with $n$ edges, and (ii) a monochromatic non-separated matching with $n$ edges. This conjecture asks whether the Cockayne–Lorimer bound remains valid for these two ordered matching types; the source identifies this as the main open problem and notes positive answers in several special cases.