arXiv subjects

William Adkisson

Publications and source records attributed to William Adkisson.

5 recordsLinked to original sources

arXiv2026

Strong Tree Properties Along Many Segments of Successors of Singulars

The strong tree property and the super tree property (also called ITP) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $\kappa$ is strongly compact if and only if the strong tree property holds at $\kappa$, and supercompact if and only if ITP holds at $\kappa$. Generalizing a result of Golshani and Hayut, we show that from large cardinals it is consistent for ITP to hold simultaneously at any countable initial segment of successors of singular cardinals. More formally, given any countable ordinal $\theta$, we construct a forcing extension in which ITP holds at the first $\theta$ successors of singulars. We then extend this result further to obtain the strong tree property on long segments of successors of singular cardinals of multiple cofinalities simultaneously.

math.LO↗

arXiv2025

On the strength of ultrafilters above choiceless large cardinals and their Prikry forcings

We study the strength of well-founded ultrafilters on ordinals above choiceless large cardinals and their associated Prikry forcings. Gabriel Goldberg showed that all but boundedly many regular cardinals above a rank Berkeley cardinal carry well-founded uniform ultrafilters. We prove several bounds on the large cardinal strength that is witnessed by such ultrafilters. We then extend the theory of Prikry forcing in this context and place limits on the cardinals that can be collapsed or singularized. Finally, we develop the notion of a tensor Prikry system, and use it to give new constructions for several consistency results in choiceless set theory. In particular, we build a new model in which all uncountable cardinals are singular.

math.LO↗

arXiv2025

Tree Properties at Successors of Singulars of Many Cofinalities

From many supercompact cardinals, we show that it is consistent for the tree property to hold at many small successors of singular cardinals, each with a different cofinality. In particular, we construct a model in which the tree property holds at $\aleph_{\omega+\omega+1}$ and at $\aleph_{\omega_n+1}$ for all $0<n<\omega$. We show that this can be done for the strong tree property as well, and extend the technique to large uncountable sequences of desired cofinalities.

math.LO↗

arXiv2022

The Strong and Super Tree Property at Successors of Singular Cardinals

The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $\kappa$ is strongly compact if and only if the strong tree property holds at $\kappa$, and supercompact if and only if ITP holds at $\kappa$. We present several results motivated by the problem of obtaining the strong tree property and ITP at many successive cardinals simultaneously; these results focus on the successors of singular cardinals. We describe a general class of forcings that will obtain the strong tree property and ITP at the successor of a singular cardinal of any cofinality. Generalizing a result of Neeman about the tree property, we show that it is consistent for ITP to hold at $\aleph_n$ for all $2 \leq n < \omega$ simultaneously with the strong tree property at $\aleph_{\omega+1}$; we also show that it is consistent for ITP to hold at $\aleph_n$ for all $3 < n < \omega$ and at $\aleph_{\omega+1}$ simultaneously. Finally, turning our attention to singular cardinals of uncountable cofinality, we show that it is consistent for the strong and super tree properties to hold at successors of singulars of multiple cofinalities simultaneously.

math.LO↗

arXiv2017

DNA Origami and Unknotted A-trails in Torus Graphs

Motivated by the problem of determining unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine existence and knottedness of A-trails in graphs embedded on the torus. We show that any A-trail in a checkerboard-colorable torus graph is unknotted and characterize the existence of A-trails in checkerboard-colorable torus graphs in terms of pairs of quasitrees in associated embeddings. Surface meshes are frequent targets for DNA nanostructure self-assembly, and so we study both triangular and rectangular torus grids. We show that, aside from one exceptional family, a triangular torus grid contains an A-trail if and only if it has an odd number of vertices, and that such an A-trail is necessarily unknotted. On the other hand, while every rectangular torus grid contains an unknotted A-trail, we also show that any torus knot can be realized as an A-trail in some rectangular grid. Lastly, we use a gluing operation to construct infinite families of triangular and rectangular grids containing unknotted A-trails on surfaces of arbitrary genus. We also give infinite families of triangular grids containing no unknotted A-trail on surfaces of arbitrary nonzero genus.

math.CO↗