Three Generalizations of Erd\H{o}s Szekeres: $k$-Modal Subsequences

Erd\H{o}s and Szekeres showed that given a permutation $p$ of $[n]$, and the sequence defined by \newline $(p(1), p(2), \ldots, p(n))$, there exists either a decreasing or increasing subsequence, not necessarily contiguous, of length at least $\sqrt{n}$. Fan Chung considered subsequences that can have at most one change of direction, i.e. an increasing and then decreasing subsequence, or a decreasing and then increasing subsequence. She called these unimodal subsequences, and showed there exists a unimodal subsequence of length at least $\sqrt{3n}$, up to some constants \cite{chung}. She conjectured that a permutation of $n$ contains a $k$-modal (at most $k$ changes in direction) subsequence of length at least $\sqrt{(2k+1)n}$ up to some constants. Zijian Xu proved this conjecture in 2024 \cite{xu}, and we will provide another substantially different proof using "sophisticated labeling arguments" instead of "underlying poset structures behind k-modal subsequences." We also show that there exists an increasing first $k$-modal subsequence of length at least $\sqrt{2kn}$.