7 problems

Two disjoint order-isomorphic subsequences of a permutation are called twins, and let $\tau(n)$ denote the maximum length of twins contained in every permutation of length $n$. Gaw…

Forbidden-pattern characterization conjecture. The ordered matching admits

Critical matching enumeration conjecture. Based on computational experiments, for $k=1$ there are exactly eight critical matchings, while for $s=q=1$ there are exactly twelve criti…

An ordered $r$-matching is an ordered matching in which every vertex belongs to one of $r$ edge labels, and let $t^{(r)}(n)$ be the minimum, over ordered $r$-matchings of size $n$,…

Let $r\geqslant 2$, let $P$ be a collectable $r$-pattern, and let $\mathbb{RM}^{(r)}_n$ denote the random ordered $r$-matching of size $n$. A $P$-clique is a set of edges whose ind…

For each integer $r\geqslant2$, let $t_r^{\operatorname{match}}(n)$ denote the largest size of twins that are unavoidable in every ordered $r$-matching of size $n$. Here twins are…

An ordered matching is a matching whose vertices are linearly ordered, and two disjoint sub-matchings are twins when they are order-isomorphic. Let $T(n)$ denote the maximum intege…