Problems in extremal graphs and poset theory

Walker, Shanise

doi:10.31274/etd-180810-6112

Cited by 2 publications

(2 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For each j ∈ N, let d * (j) be the largest integer d for which d ⌊d/2⌋ ≤ j − 1. Therefore, each gap in the chain for color class j has size at most d * A table of values of k, 1 ≤ k ≤ 300, is given in [9] which determines whether Theorem 11 (1) or Theorem 11(2) is the better lower bound for sat * (n, A k+1 ) when n is sufficiently large.…”

Section: Induced-a K+1 -Saturated Resultsmentioning

confidence: 99%

Improved bounds for induced poset saturation

Martin¹,

Smith²,

Walker³

2019

Preprint

Self Cite

View full text Add to dashboard Cite

Given a finite poset P, a family F of elements in the Boolean lattice is induced-P-saturated if F contains no copy of P as an induced subposet but every proper superset of F contains a copy of P as an induced subposet. The minimum size of an induced-P-saturated family in the n-dimensional Boolean lattice, denoted sat * (n, P), was first studied by Ferrara et al. (2017).Our work focuses on strengthening lower bounds. For the 4point poset known as the diamond, we prove sat * (n, D 2 ) ≥ √ n, improving upon a logarithmic lower bound. For the antichain with k + 1 elements, we prove sat * (n, A k+1 ) ≥ (1 − o k (1)) kn log 2 k , improving upon a lower bound of 3n − 1 for k ≥ 3.

show abstract

Section: Induced-a K+1 -Saturated Resultsmentioning

confidence: 99%

Improved bounds for induced poset saturation

Martin¹,

Smith²,

Walker³

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…For a gap (X, Y ) which has size d, note that (X, Y ) ∪ {X, Y } is isomorphic to B d . Sperner's theorem [8] gives that the maximum size of an antichain in B d is A table of values of k, 1 k 300, is given in [9] which determines whether Theorem 11 (1) or Theorem 11(2) is the better lower bound for sat * (n, A k+1 ) when n is sufficiently large.…”

Section: Introductionmentioning

confidence: 99%

Improved Bounds for Induced Poset Saturation

Martin

Smith

Walker

2020

Electron. J. Combin.

View full text Add to dashboard Cite

Given a finite poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the Boolean lattice is induced-$\mathcal{P}$-saturated if $\mathcal{F}$ contains no copy of $\mathcal{P}$ as an induced subposet but every proper superset of $\mathcal{F}$ contains a copy of $\mathcal{P}$ as an induced subposet. The minimum size of an induced-$\mathcal{P}$-saturated family in the $n$-dimensional Boolean lattice, denoted $\mathrm{sat}^*(n,\mathcal{P})$, was first studied by Ferrara et al. (2017). Our work focuses on strengthening lower bounds. For the 4-point poset known as the diamond, we prove $\mathrm{sat}^*(n,\Diamond)\geq\sqrt{n}$, improving upon a logarithmic lower bound. For the antichain with $k+1$ elements, we prove $$\mathrm{sat}^*(n,\mathcal{A}_{k+1})\geq \left(1-\frac{1}{\log_2k}\right)\frac{kn}{\log_2 k}$$ for $n$ sufficiently large, improving upon a lower bound of $3n-1$ for $k\geq 3$.

show abstract

Problems in extremal graphs and poset theory

Cited by 2 publications

References 30 publications

Improved bounds for induced poset saturation

Improved bounds for induced poset saturation

Improved Bounds for Induced Poset Saturation

Contact Info

Product

Resources

About