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Induced saturation for complete bipartite posets

Dingyuan Liu

TL;DR

This paper resolves the asymptotic size of induced $\\mathcal{K}_{s,t}$-saturated families in the Boolean lattice by establishing a linear upper bound for fixed $s,t\ge 2$ and a linear lower bound for a broad class of posets. The authors introduce a lantern-based construction to build an induced $\\mathcal{K}_{s,t}$-free family of size $O(n)$ and then append a final layer to achieve saturation, proving $\\mathrm{sat}^{*}(n,\\mathcal{K}_{s,t}) = O(n)$. They also prove a general $n+1$ lower bound for any poset with legs, which includes $\\mathcal{K}_{s,2}$, thereby showing $\\mathrm{sat}^{*}(n,\\mathcal{K}_{s,2}) = \Theta(n)$ and supporting the linear regime for a wide class of posets. Together, these results refute expectations of universal superlinear growth for induced saturation in these families and contribute to the understanding of the induced saturation dichotomy in the Boolean lattice.

Abstract

Given $s,t\in\mathbb{N}$, a complete bipartite poset $\mathcal{K}_{s,t}$ is a poset whose Hasse diagram consists of $s$ pairwise incomparable vertices in the upper layer and $t$ pairwise incomparable vertices in the lower layer, such that every vertex in the upper layer is larger than all vertices in the lower layer. A family $\mathcal{F}\subseteq2^{[n]}$ is called induced $\mathcal{K}_{s,t}$-saturated if $(\mathcal{F},\subseteq)$ contains no induced copy of $\mathcal{K}_{s,t}$, whereas adding any set from $2^{[n]}\backslash\mathcal{F}$ to $\mathcal{F}$ creates an induced $\mathcal{K}_{s,t}$. Let $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})$ denote the smallest size of an induced $\mathcal{K}_{s,t}$-saturated family $\mathcal{F}\subseteq2^{[n]}$. It was conjectured that $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})$ is superlinear in $n$ for certain values of $s$ and $t$. In this paper, we show that $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})=O(n)$ for all fixed $s,t\in\mathbb{N}$. Moreover, we prove a linear lower bound on $\mathrm{sat}^{*}(n,\mathcal{P})$ for a large class of posets $\mathcal{P}$, particularly for $\mathcal{K}_{s,2}$ with $s\in\mathbb{N}$.

Induced saturation for complete bipartite posets

TL;DR

This paper resolves the asymptotic size of induced -saturated families in the Boolean lattice by establishing a linear upper bound for fixed and a linear lower bound for a broad class of posets. The authors introduce a lantern-based construction to build an induced -free family of size and then append a final layer to achieve saturation, proving . They also prove a general lower bound for any poset with legs, which includes , thereby showing and supporting the linear regime for a wide class of posets. Together, these results refute expectations of universal superlinear growth for induced saturation in these families and contribute to the understanding of the induced saturation dichotomy in the Boolean lattice.

Abstract

Given , a complete bipartite poset is a poset whose Hasse diagram consists of pairwise incomparable vertices in the upper layer and pairwise incomparable vertices in the lower layer, such that every vertex in the upper layer is larger than all vertices in the lower layer. A family is called induced -saturated if contains no induced copy of , whereas adding any set from to creates an induced . Let denote the smallest size of an induced -saturated family . It was conjectured that is superlinear in for certain values of and . In this paper, we show that for all fixed . Moreover, we prove a linear lower bound on for a large class of posets , particularly for with .
Paper Structure (6 sections, 13 theorems, 8 equations, 9 figures)

This paper contains 6 sections, 13 theorems, 8 equations, 9 figures.

Table of Contents

  1. Introduction
  2. A linear upper bound for $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})$
  3. Construction of an induced $\mathcal{K}_{s,t}$-free family
  4. Induced $\mathcal{K}_{s,t}$-saturated
  5. Proof of Theorem \ref{['thm1']}
  6. Proof of Theorem \ref{['thm3']}

Key Result

Theorem 1.3

Let $n,s,t\in\mathbb{N}$ with $s\geq{t}\geq2$ and $n\geq{2s+t-1}$. Then where $c_{s,t}$ is some constant depending on $s$ and $t$.

Figures (9)

  • Figure 1: The Hasse diagrams of $\mathcal{K}_{1,1}$, $\mathcal{K}_{2,1}$, $\mathcal{K}_{3,1}$, and $\mathcal{K}_{2,2}$.
  • Figure 2: $\mathcal{L}^{s}(A)$ and $\mathcal{L}_{t}(A)$.
  • Figure 3: The hypothetical induced copy of $\mathcal{K}_{s,t}$.
  • Figure 4: The induced copy of $\mathcal{K}_{s,t}$ in Case 1.
  • Figure 5: The induced copy of $\mathcal{K}_{s,t}$ in Case 2.
  • ...and 4 more figures

Theorems & Definitions (31)

  • Conjecture 1.1: keszegh2021induced
  • Conjecture 1.2: ivan2020saturation
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Remark 1
  • ...and 21 more