Every Large Point Set contains Many Collinear Points or an Empty Pentagon (2009)
Abel, Zachary, Ballinger, Brad, Bose, Prosenjit, Collette, Sébastien, Dujmović, Vida, Hurtado, Ferran, ...
We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon....
Cauchy's Arm Lemma on a Growing Sphere (2008)
Abel, Zachary, Charlton, David, Collette, Sebastien, Demaine, Erik D., Demaine, Martin L., Langerman, Stefan, ...
We propose a variant of Cauchy's Lemma, proving that when a convex chain on one sphere is redrawn (with the same lengths and angles) on a larger sphere, the distance between its endpoints increases....
Pushing Hypercubes Around (2008)
Abel, Zachary, Kominers, Scott D.
We fully generalize the result of Dumitrescu and Pach concerning two-dimensional modular metamorphic systems. In particular, we show that for any two connected n-module configurations V and V' of...
Hinged Dissections Exist (2007)
Abbott, Timothy G., Abel, Zachary, Charlton, David, Demaine, Erik D., Demaine, Martin L., Kominers, Scott D.
We prove that any finite collection of polygons of equal area has a common hinged dissection. That is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can...
Configurations of Rank-40r Extremal Even Unimodular Lattices (r=1,2,3) (2007)
Kominers, Scott D., Abel, Zachary
We show that if L is an extremal even unimodular lattice of rank 40r with r=1,2,3 then L is generated by its vectors of norms 4r and 4r+2. Our result is an extension of Ozeki's result for the case...
A Categorical Construction of Ultrafilters (2007)
Litt, Daniel, Abel, Zachary, Kominers, Scott D.
We provide a brief exposition of ultrafilters and offer a categorical construction for them in terms of the inverse limit of an inverse family of finite partitions. From this construction, we derive...