arXiv subjects
Publications and source records attributed to Zach Hunter.
A corner is a triple of points in $\Bbb{Z}^2$ of the form $(x,y),(x+d,y),(x,y+d)$ where $d\neq 0$. One can think of them as being 2D-analogues to 3-term arithmetic progressions. In this short note, we extend ideas of Green-Wolf from this latter setting to the former, achieving slightly better constructions of corner-free sets.
Recently, Ben Green proved that the two-color van der Waerden number $w(3,k)$ is bounded from below by $k^{b_0(k)}$ where $b_0(k) = c_0\left(\frac{\log k }{\log \log k}\right)^{1/3}$. We prove a new lower bound of $k^{b(k)}$ with $b(k) = \frac{c\log k}{\log \log k}$. This is done by modifying Green's argument, replacing a complicated result about random quadratic forms with an elementary probabilistic result.
It was proved by Scott that for every $k\ge2$, there exists a constant $c(k)>0$ such that for every bipartite $n$-vertex graph $G$ without isolated vertices, there exists an induced subgraph $H$ of order at least $c(k)n$ such that $\textrm{deg}_H(v) \equiv 1\pmod{k}$ for each $v \in H$. Scott conjectured that $c(k) = Ω(1/k)$, which would be tight up to the multiplicative constant. We confirm this conjecture.
For a graph $G$, the $\ell$-deck of $G$ is the multiset of induced subgraphs on $G$ having $\ell$ vertices. Recently, Groenland et al. proved that any tree can be reconstructed from its $(8/9+o(1))n$-deck. For the particular case of caterpillar graphs, we show that the $(1/2+o(1))n$-deck suffices, which is asymptotically tight.
A permutation $σ\in S_n$ is a $k$-superpattern (or $k$-universal) if it contains each $τ\in S_k$ as a pattern. This notion of "superpatterns" can be generalized to words on smaller alphabets, and several questions about superpatterns on small alphabets have recently been raised in the survey of Engen and Vatter. One of these questions concerned the length of the shortest $k$-superpattern on $[k+1]$. A construction by Miller gave an upper bound of $(k^2+k)/2$, which we show is optimal up to lower-order terms. This implies a weaker version of a conjecture by Eriksson, Eriksson, Linusson and Wastlund. Our results also refute a 40-year-old conjecture of Gupta.
We develop a new Markov chain on graph partitions that makes relatively global moves yet is computationally feasible to be used as the proposal in the Metropolis-Hastings method. Our resulting algorithm can be made reversible and able to sample from a specified measure on partitions. Both of these properties are critical to some important applications and computational Bayesian statistics in general. Our proposal chain modifies the recently developed method called Recombination (ReCom), which draws spanning trees on joined partitions and then randomly cuts them to repartition. We improve the computational efficiency by augmenting the state space from partitions to spanning forests. The extra information accelerates the computation of the forward and backward proposal probabilities. We demonstrate this method by sampling redistricting plans and find promising convergence results on several key observables of interest.
We develop a Multi-Scale Merge-Split Markov chain on redistricting plans. The chain is designed to be usable as the proposal in a Markov Chain Monte Carlo (MCMC) algorithm. Sampling the space of plans amounts to dividing a graph into a partition with a specified number of elements which each correspond to a different district. The districts satisfy a collection of hard constraints and the measure may be weighted with regard to a number of other criteria. The multi-scale algorithm is similar to our previously developed Merge-Split proposal, however, this algorithm provides improved scaling properties and may also be used to preserve nested communities of interest such as counties and precincts. Both works use a proposal which extends the ReCom algorithm which leveraged spanning trees merge and split districts. In this work we extend the state space so that each district is defined by a hierarchy of trees. In this sense, the proposal step in both algorithms can be seen as a "Forest ReCom." We also expand the state space to include edges that link specified districts, which further improves the computational efficiency of our algorithm. The collection of plans sampled by the MCMC algorithm can serve as a baseline against which a particular plan of interest is compared. If a given plan has different racial or partisan qualities than what is typical of the collection of plans, the given plan may have been gerrymandered and is labeled as an outlier.
North Carolina's constitution requires that state legislative districts should not split counties. However, counties must be split to comply with the "one person, one vote" mandate of the U.S. Supreme Court. Given that counties must be split, the North Carolina legislature and courts have provided guidelines that seek to reduce counties split across districts while also complying with the "one person, one vote" criteria. Under these guidelines, the counties are separated into clusters. The primary goal of this work is to develop, present, and publicly release an algorithm to optimally cluster counties according to the guidelines set by the court in 2015. We use this tool to investigate the optimality and uniqueness of the enacted clusters under the 2017 redistricting process. We verify that the enacted clusters are optimal, but find other optimal choices. We emphasize that the tool we provide lists \textit{all} possible optimal county clusterings. We also explore the stability of clustering under changing statewide populations and project what the county clusters may look like in the next redistricting cycle beginning in 2020/2021.