PhD Dissertation Defense: "Continuous combinatorics on $F(2^(Z^2) )$" | Department of Mathematics

COVID-19 updates—In an effort to keep everyone healthy, UNT's on-campus operations are closed until further notice. We're serving students remotely. Please stay connected. Stay up to date on UNT’s response to COVID-19 (Coronavirus).

Have you registered for fall classes yet? While COVID-19 has created some uncertainty for us all, UNT is committed to helping the Mean Green family turn dreams into reality. Let's get through this together!

Register for classes on
Not a UNT student yet? Apply to UNT
Having trouble registering? Get help from an advisor

PhD Dissertation Defense: "Continuous combinatorics on $F(2^(Z^2) )$"

Event Information
Event Location: 
GAB 461
Event Date: 
Tuesday, March 22, 2016 - 3:00pm

Professor Steve Jackson invites you to attend the PhD dissertation defense of Edward Krohne today, March 22nd at 3:00 pm in GAB 461. Cake and coffee will be served in GAB 472 following this event.

"Continuous combinatorics on $F(2^(Z^2 ) )$"


We present a novel theorem of Borel Combinatorics that sheds light on the types of continuous functions that can be defined on the graph of $F(2^(Z^2 ) )$. The topological space $F(2^(Z^2 ) )$ embeds into the Cantor space $2^ω$ and has a natural free continuous $Z^2$ action. Considering the graph induced by this action, we obtain a disjoint union of uncountably many Cayley graphs of $Z^2$. It is folklore that no continuous (indeed, Borel) function provides a chromatic two-coloring of $F(2^(Z^2 ) )$, despite the fact that any finite part of the graph on $F(2^(Z^2 ) )$ is bipartite. Our main result offers a much more complete analysis of continuous functions on this space. We construct a countable collection of finite graphs $Γ_(n,p,q)$ (with n≤p⊥q≤ω) each consisting of twelve "tiles," such that for any property P (such as "chromatic two-coloring") that is locally recognizable in the proper sense, a continuous function with property P exists on $F(2^(Z^2 ) )$ iff a function with a corresponding property $P_0$ exists on some $Γ_(n,p,q)$. We present the theorem, and give several applications.

Thinking about UNT?

It's easy to apply online. Join us and discover why we're the choice of over 38,000 students.

Apply now