## Date of Award

Summer 2022

## Document Type

Open Access Dissertation

## Department

Mathematics

## First Advisor

Linyuan Lu

## Abstract

For each positive integer *n*, let Q*n* denote the Boolean lattice of dimension *n*. For posets *P*, *P'*, define the *poset Ramsey number R*(*P*,*P'*) to be the least *N* such that for any red/blue coloring of the elements of Q_{N}, there exists either a subposet isomorphic to *P* with all elements red, or a subposet isomorphic to *P'* with all elements blue.

Axenovich and Walzer introduced this concept in *Order* (2017), where they proved *R*(Q_{2}, Q_{n}) ≤* 2n* + 2 and *R*(Q* _{n}*, Q

_{m}) ≤

*mn*+

*n*+

*m*. They later proved 2

*n*≤

*R*(Q

_{n}, Q

_{n}) ≤ n

_{2}+ 2

*n*. Walzer later proved

*R*(Q

*, Q*

_{n}*) ≤*

_{n}*n*

_{2}+ 1. We provide some improved bounds for

*R*(Q

_{n}, Q

_{m}) for various

*n*,

*m ∈ u+2115*. In particular, we prove that

*R*(

*Qn*,

*Qn*) ≤

*n*

^{2}-

*n*+ 2,

*R*(

*Q*

_{2},

*Q*

_{n}) ≤ 5/3

*n*+ 2, and

*R*(

*Q*

_{3},

*Q*

_{n}) ≤ ⌈ 37/16

*n*+ 55/16 ⌉. We also prove that

*R*(Q

_{2},Q

_{3}) = 5, and R(Q

*, Q*

_{m}*) ≤ ⌈(*

_{n}*m*- 1 + 2/(

*m*+1))

*n*+ 1/3

*m*+ 2 ⌉ for all

*n*>

*m*≥ 4.

## Rights

© 2022, Joshua Cain Thompson

## Recommended Citation

Thompson, J. C.(2022). *Poset Ramsey Numbers for Boolean Lattices.* (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/6931