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Largest component in Boolean sublattices

Julian Galliano, Ross J. Kang

TL;DR

This work investigates how large a subfamily F⊆2^{[n]} can be before the comparability graph G_F gains a component larger than t, generalising Sperner's threshold at La(n). The authors relate the problem to the Lubell function and establish that for t=2^{g(n)} with g(n)=o(n/log n), La(n,t)=(1+o(1))La(n), via a reduction to a rainbow-cycle Turán-type problem and leveraging the ABSZZ+ bound D^*(t,O^*)=O(log t log log t). They also obtain sharp bounds for the largest disconnected family and present exact results for special cases k=1 and k=2, including a diamond-based generalisation of the BLYM inequality. The results reveal a condensation phenomenon near the Sperner threshold and connect extremal lattice theory with rainbow-cycle Turán-type problems, offering both exact and asymptotic insights and motivating further questions about the strength of these bounds and their extensions.

Abstract

For a subfamily ${F}\subseteq 2^{[n]}$ of the Boolean lattice, consider the graph $G_{F}$ on ${F}$ based on the pairwise inclusion relations among its members. Given a positive integer $t$, how large can ${F}$ be before $G_{F}$ must contain some component of order greater than $t$? For $t=1$, this question was answered exactly almost a century ago by Sperner: the size of a middle layer of the Boolean lattice. For $t=2^n$, this question is trivial. We are interested in what happens between these two extremes. For $t=2^{g}$ with $g=g(n)$ being any integer function that satisfies $g(n)=o(n/\log n)$ as $n\to\infty$, we give an asymptotically sharp answer to the above question: not much larger than the size of a middle layer. This constitutes a nontrivial generalisation of Sperner's theorem. We do so by a reduction to a Turán-type problem for rainbow cycles in properly edge-coloured graphs. Among other results, we also give a sharp answer to the question, how large can ${F}$ be before $G_{F}$ must be connected?

Largest component in Boolean sublattices

TL;DR

This work investigates how large a subfamily F⊆2^{[n]} can be before the comparability graph G_F gains a component larger than t, generalising Sperner's threshold at La(n). The authors relate the problem to the Lubell function and establish that for t=2^{g(n)} with g(n)=o(n/log n), La(n,t)=(1+o(1))La(n), via a reduction to a rainbow-cycle Turán-type problem and leveraging the ABSZZ+ bound D^*(t,O^*)=O(log t log log t). They also obtain sharp bounds for the largest disconnected family and present exact results for special cases k=1 and k=2, including a diamond-based generalisation of the BLYM inequality. The results reveal a condensation phenomenon near the Sperner threshold and connect extremal lattice theory with rainbow-cycle Turán-type problems, offering both exact and asymptotic insights and motivating further questions about the strength of these bounds and their extensions.

Abstract

For a subfamily of the Boolean lattice, consider the graph on based on the pairwise inclusion relations among its members. Given a positive integer , how large can be before must contain some component of order greater than ? For , this question was answered exactly almost a century ago by Sperner: the size of a middle layer of the Boolean lattice. For , this question is trivial. We are interested in what happens between these two extremes. For with being any integer function that satisfies as , we give an asymptotically sharp answer to the above question: not much larger than the size of a middle layer. This constitutes a nontrivial generalisation of Sperner's theorem. We do so by a reduction to a Turán-type problem for rainbow cycles in properly edge-coloured graphs. Among other results, we also give a sharp answer to the question, how large can be before must be connected?

Paper Structure

This paper contains 11 sections, 20 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Proof of \ref{['main']}
  4. The Lubell function
  5. Towards \ref{['main']}
  6. Bounding $\xi^\star_n$
  7. Exact approaches
  8. Generalising the BLYM inequality
  9. The cases $k=1$ and $k=2$
  10. Largest disconnected family
  11. Concluding remarks

Key Result

Proposition 1.2

For all integers $k,n$ with $0\le k\le n$, there exists a family $\mathcal{F}\subseteq 2^{[n]}$ of size $2^k\binom{n-k}{\lfloor (n-k)/2\rfloor}$ such that the components of $G_\mathcal{F}$ have order $2^k$.

Figures (2)

  • Figure 1: $S(C_k)$ is split into $|C_k|k(n-k)$ parts, three of which are depicted here. The leftmost part contains only good permutations, the other two contain only bad permutations.
  • Figure 2: The sets $\mathcal{F}^+$ and $\mathcal{F}^-$.

Theorems & Definitions (45)

  • Conjecture 1.1
  • Proposition 1.2
  • proof
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: Kru63Kat68
  • Theorem 1.6: Lub66Bol65Yam54Mes63
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 35 more