Minimum number of monotone subsequences of length 4 in permutations

TL;DR

The paper establishes a lower bound on the number of monotone subsequences of length four in large permutations and characterizes the permutations that achieve this bound, using flag algebra methods.

Contribution

It introduces a new lower bound for monotone subsequences of length four and characterizes extremal permutations that attain this bound.

Findings

Lower bound on monotone 4-subsequences in permutations.

Characterization of extremal permutations with minimal monotone 4-subsequences.

Connection to edge colorings of complete graphs minimizing monochromatic K4s.

Abstract

We show that for every sufficiently large $n$, the number of monotone subsequences of length four in a permutation on $n$ points is at least $\binom{\lfloor n/3 \rfloor}{4} + \binom{\lfloor(n+1)/3\rfloor}{4} + \binom{\lfloor (n+2)/3\rfloor}{4}$. Furthermore, we characterize all permutations on $[n]$ that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colorings of complete graphs with two colors, where the number of monochromatic $K_4$'s is minimized. We show that all the extremal colorings must contain monochromatic $K_4$'s only in one of the two colors. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.