[Paper Review] Linear orderings of combinatorial cubes

Boris Bukh, Anish Sevekari|arXiv (Cornell University)|Jun 27, 2019

graph theory and CDMA systems8 references4 citations

TL;DR

This paper establishes that every linear ordering of the Boolean cube $[2]^n$ contains a large subcube where the ordering is lexicographic, and more generally, every linear ordering of $[k]^n$ contains a large subcube where the ordering agrees with one of approximately $\frac{(k-1)!}{2(\ln 2)^k}$ canonical orderings. The key contribution is a Hales–Jewett-type Ramsey result for combinatorial cubes, implying that long sequences contain monotone subsequences supported on affine cubes.

ABSTRACT

We show that, for every linear ordering of $[2]^n$, there is a large subcube on which the ordering is lexicographic. We use this to deduce that every long sequence contains a long monotone subsequence supported on an affine cube. More generally, we prove an analogous result for linear orderings of $[k]^n$. We show that, for every such ordering, there is a large subcube on which the ordering agrees with one of approximately $\frac{(k-1)!}{2(\ln 2)^k}$ orderings.

Motivation & Objective

• To extend the Erdős–Szekeres theorem by requiring monotone subsequences to be supported on affine cubes rather than just index sets.
• To establish a Hales–Jewett-type Ramsey result for linear orderings on $[2]^n$, showing that every such ordering contains a lexicographically ordered subcube.
• To generalize this result to $[k]^n$ for arbitrary $k \geq 2$, identifying a bounded number of canonical orderings that must appear in any linear ordering.
• To apply these results to deduce the existence of long monotone subsequences with indices forming affine cubes in arbitrary real sequences.
• To explore the quantitative dependence of subcube size on dimension, particularly the conjectured doubly exponential growth in the number of elements required.

Proposed method

• Uses canonical parameter words to define $d$-subcubes of $[k]^n$, enabling consistent identification with $[k]^d$ via bijection.
• Defines a restriction of a linear ordering on $[k]^n$ to a subcube as a linear ordering on $[k]^d$ via the canonical bijection.
• Applies the Graham–Rothschild theorem to color pairs of words in $[2]^n$ based on a coloring of $[m]$, with $m = 3^n$, to find monochromatic configurations.
• Introduces the concept of incomparable words (with both $0,1$ and $1,0$ pairs) to identify sets whose images under projection form affine cubes.
• Uses induction and uniformity arguments to show that if two words are ordered in a certain way, their extensions remain ordered under the same rule.
• Leverages the structure of $d$-subcubes and canonical words to reduce the problem to known Ramsey-theoretic results on high-dimensional cubes.

Experimental results

Research questions

• RQ1 Can every long sequence contain a monotone subsequence whose index set is a proper affine $d$-cube?
• RQ2 Does every linear ordering of $[2]^n$ contain a large subcube on which the ordering is lexicographic?
• RQ3 For $[k]^n$, is there a bounded number of canonical orderings such that every linear ordering contains a large subcube agreeing with one of them?
• RQ4 Can the Erdős–Szekeres theorem be strengthened to require monotone subsequences supported on combinatorial cubes?
• RQ5 What is the optimal quantitative dependence of the required sequence length on $d$ in the affine cube setting?

Key findings

• For every $d$, there exists $m$ such that every sequence of $m$ distinct reals contains a monotone subsequence whose index set is a proper affine $d$-cube.
• Every linear ordering of $[2]^n$ contains a $d$-subcube on which the ordering is lexicographic for one of the two linear orderings of $[2]$.
• For $[k]^n$, every linear ordering contains a large subcube where the restriction agrees with one of approximately $\frac{(k-1)!}{2(\ln 2)^k}$ canonical orderings.
• The existence of such subcubes implies that every long sequence contains a monotone subsequence supported on an affine cube.
• The dependence of $m$ on $d$ in Theorem 1 is likely doubly exponential, though the paper does not optimize this bound.
• The proof uses the Graham–Rothschild theorem to find monochromatic $2d$-subcubes, which are then used to extract monochromatic affine cubes in the index set.