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On $k$-modal subsequences

Zijian Xu

Abstract

A $k$-modal sequence is a sequence of real numbers that can be partitioned into $k+1$ (possibly empty) monotone sections such that adjacent sections have opposite monotonicities. For every positive integer $k$, we prove that any sequence of $n$ pairwise distinct real numbers contains a $k$-modal subsequence of length at least $\sqrt{(2k+1)(n-\frac14)} - \frac{k}{2}$, which is tight in a strong sense. This confirms an old conjecture of F.R.K.Chung (J.Combin.Theory Ser.A, 29(3):267-279, 1980).

On $k$-modal subsequences

Abstract

A -modal sequence is a sequence of real numbers that can be partitioned into (possibly empty) monotone sections such that adjacent sections have opposite monotonicities. For every positive integer , we prove that any sequence of pairwise distinct real numbers contains a -modal subsequence of length at least , which is tight in a strong sense. This confirms an old conjecture of F.R.K.Chung (J.Combin.Theory Ser.A, 29(3):267-279, 1980).
Paper Structure (5 sections, 14 theorems, 33 equations)

This paper contains 5 sections, 14 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Modal paths and fine coverings
  3. Proof of Theorem \ref{['thm:fine_covering']}
  4. Generic sets without long k-modal paths
  5. Proof of Lemma \ref{['lem:inj']}

Key Result

Theorem 1

For any integers $k \ge 0$ and $n \ge 1$, we have $\rho(n; k) \ge \Bigl\lceil \sqrt{(2k+1)(n-\frac{1}{4})} - \frac{k}{2} \Bigr\rceil$.

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 5
  • Proposition 6
  • proof
  • Theorem 7
  • proof : Proof of \ref{['thm:modal_subpth']} assuming \ref{['thm:fine_covering']}
  • proof
  • ...and 17 more