Neal Bushaw

Overview

Teaching

Courses this semester

Email

nobushaw@vcu.edu

Office

Grace E. Harris Hall 4107

Mail

Department of Mathematics and Statistics
Box 842014, 1015 Floyd Avenue
Richmond, Virginia 23284

Research

My interests lie primarily within combinatorics, the area of mathematics which studies finite discrete structures. Recently, my research has focused on new aspects of the forbidden subgraph problem. However, I am also very interested in the use of probability in deterministic settings, as well as a wide variety of probabilistic and extremal questions on sets, graphs, and hypergraphs.

Publications and preprints

• N. Bushaw and A. Clifton. 3-neighbor bootstrap percolation on two-dimensional grids. Preprint. arXiv
  Abstract

  In the $3$-neighbor bootstrap percolation process, a vertex becomes (and remains) infected if at least three of its neighbors are infected. We say that an initial configuration of infected vertices percolates if eventually all vertices are infected. We exactly determine the size of the minimum percolating set for the $3$-neighbor bootstrap percolation process on all remaining open cases for rectangular grid graphs $P_m\mathrm{\square }{P}_n$. This extends earlier work of Dukes, Noel, and Romer. Additionally, we consider the same question for the toroidal grids $C_m\mathrm{\square }{C}_n$, proving upper and lower bounds which are at most one apart and determining the answer precisely in many divisibility cases.

• N. Bushaw. Cyclic and Möbius pianos. Bridges 2026 Conference Proceedings, 591–594. Bridges archive
  Abstract

  The intent of this article is to unify several distinct ways of thinking about piano keyboards. Instead of treating the piano as a long line of keys, we discuss keyboards which repeat after a small number of steps in either direction – that is, we think of a keyboard as being built from repeating octaves. On a standard piano, the black keys in each of these octaves are spread as evenly as possible among the white keys; this phenomenon has been explained in several different mathematical manners. Our new cyclic pianos have black keys which are minimum energy sets on cycles and Möbius ladders. We discuss the mathematics of such pianos, and provide a software tool for simultaneously exploring the music and mathematics of arbitrary cyclic and Möbius pianos.

• N. Bushaw, J. Danielsson and G. Hurlbert. Erdős–Ko–Rado theorems for paths in graphs. Submitted. arXiv
  Abstract

  A family of sets is $s$-intersecting if every pair of its sets has at least $s$ elements in common. It is an $s$-star if all its members have some $s$ elements in common. A family of sets is called $s$-EKR if all its $s$-intersecting subfamilies have size at most that of some $s$-star. For example, the classic 1961 Erdős-Ko-Rado theorem states essentially that the family of $r$-sized subsets of $\{1,2,\dots ,n\}$ is $s$-EKR when $n$ is a large enough function of $r$ and $s$, and the 1967 Hilton-Milner theorem provides the near-star structure of the largest non-star intersecting family of such sets. Two important conjectures along these lines followed: by Chvátal in 1974, that every subset-closed family of sets is 1-EKR, and by Holroyd and Talbot in 2005, that, for every graph, the family of all its $r$-sized independent sets is 1-EKR when every maximal independent set has size at least $2r$.

  In this paper we present similar 1-EKR results for families of length-$r$ paths in graphs, specifically for sun graphs, which are cycles with pendant edges attached in a uniform way, and theta graphs, which are collections of pairwise internally disjoint paths sharing the same two endpoints. We also prove $s$-EKR results for such paths in suns, and give a Hilton-Milner type result for them as well. A set is a transversal of a family of sets if it intersects each member of the family, and the transversal number of the family is the size of its smallest transversal. For example, stars have transversal number 1, and the Hilton-Milner family has transversal number 2. We conclude the paper with some transversal results involving what we call triangular families, including a few results for projective planes.

• V. Bardenova, N. Bushaw, B. Cody, P. Fay and M. Tennant. The Wiener index of vertex colorings. Submitted. arXiv
  Abstract

  The Wiener index of a vertex coloring of a graph is defined to be the sum of all pairwise geodesic distances between vertices of the same color. We provide characterizations of vertex colorings of paths and cycles whose Wiener index is as large as possible over various natural collections. Along the way we establish a connection between the majorization order on tuples of integers and the Wiener index of vertex colorings on paths and cycles.

• N. Bushaw, S. English, E. Heath, D. P. Johnston and P. Rombach. The saturation spectrum of Berge stars. Submitted. arXiv
  Abstract

  The forbidden subgraph problem is among the oldest in extremal combinatorics -- how many edges can an $n$-vertex $F$-free graph have? The answer to this question is the well-studied extremal number of $F$. Observing that every extremal example must be maximally $F$-free, a natural minimization problem is also studied -- how few edges can an $n$-vertex maximal $F$-free graph have? This leads to the saturation number of $F$. Both of these problems are notoriously difficult to extend to $k$-uniform hypergraphs for any $k\ge 3$.

  Barefoot et al., in the case of forbidding triangles in graphs, asked a beautiful question -- which numbers of edges, between the saturation number and the extremal number, are actually realized by an $n$-vertex maximal $F$-free graph? Hence named the saturation spectrum of $F$, this has since been determined precisely for several classes of graphs through a large number of papers over the past two decades.

  In this paper, we extend the notion of the saturation spectrum to the hypergraph context. Given a graph $F$ and a hypergraph $G$ embedded on the same vertex set, we say $G$ is a Berge-$F$ if there exists a bijection $\varphi :E(F)\to E(G)$ such that $e\subseteq \varphi (e)$ for all $e\in E(F)$. We completely determine the saturation spectrum for $3$-uniform Berge-$K_{1,\mathrm{\ell }}$ for $1\le \mathrm{\ell }\le 4$, and for $\mathrm{\ell }=5$ when $5\mid n$. We also determine all but a constant number of values in the spectrum for $3$-uniform Berge-$K_{1,\mathrm{\ell }}$ for all $\mathrm{\ell }\ge 5$. We note that this is the first result determining the saturation spectrum for any non-trivial hypergraph.

• V. Bednar and N. Bushaw. Rainbow Turán methods for trees. Australasian Journal of Combinatorics 91 (2025), 266–281. Australasian J. Combinatorics (PDF) · arXiv
  Abstract

  The rainbow Turán number, a natural extension of the well studied traditional Turán number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstraëte. The rainbow Turán number of a graph $H$, $e{x}^{\ast }(n,H)$, is the largest number of edges for an $n$ vertex graph $G$ which can be properly edge colored with no rainbow $H$ subgraph.

  We explore the reduction method for finding upper bounds on rainbow Turán numbers, and use this to inform results for the rainbow Turán numbers of double stars, caterpillars, and perfect binary trees. In addition, we define $k$-unique colorings and the related $k$-unique Turán numbers. We provide preliminary results on this new variant on the classic problem.

• N. Bushaw, B. Cody and C. Leffler. Sets of vertices with extremal energy. Discrete Mathematics 348 (8) (2025), 114466. Circulated in earlier versions as Maximal evenness in graphs . arXiv
  Abstract

  We define various notions of energy of a set of vertices in a graph, which generalize two of the most widely studied graphical indices: the Wiener index and the Harary index. We provide a new proof of a result due to Douthett and Krantz, which says that for cycles, the sets of vertices which have minimal energy among all sets of the same size are precisely the maximally even sets, as defined in Clough and Douthett's work on music theory. Generalizing a theorem of Clough and Douthett, we prove that a finite, simple, connected graph is distance degree regular if and only if whenever a set of vertices has minimal energy, its complement also has minimal energy. We also provide several characterizations of sets of vertices in finite paths and cycles for which the sum of all pairwise distances between vertices in the set is maximal among all sets of the same size.

• N. Bushaw and N. Kettle. Thresholds for pebbling on grids. Discrete Mathematics 348 (10) (2025), 114519. arXiv
  Abstract

  Given a connected graph $G$ and a configuration of $t$ pebbles on the vertices of G, a $q$-pebbling step consists of removing $q$ pebbles from a vertex, and adding a single pebble to one of its neighbors. Given a vector $\mathbf{q}=({\mathbf{q}}_{\mathbf{1}},\dots ,{\mathbf{q}}_{\mathbf{d}})$, $\mathbf{q}$-pebbling consists of allowing $q_i$-pebbling in coordinate $i$. A distribution of pebbles is called solvable if it is possible to transfer at least one pebble to any specified vertex of $G$ via a finite sequence of pebbling steps.

  In this paper, we determine the weak threshold for $\mathbf{q}$-pebbling on the sequence of grids $[n{]}^d$ for fixed $d$ and $\mathbf{q}$, as $n\to \mathrm{\infty }$. Further, we determine the strong threshold for $q$-pebbling on the sequence of paths of increasing length. A fundamental tool in these proofs is a new notion of centrality, and a sufficient condition for solvability based on the well used pebbling weight functions; we believe this weight lemma to be the first result of its kind, and may be of independent interest.

  These theorems improve recent results of Czygrinow and Hurlbert, and Godbole, Jablonski, Salzman, and Wierman. They are the generalizations to the random setting of much earlier results of Chung.

  In addition, we give a short counterexample showing that the threshold version of a well known conjecture of Graham does not hold. This uses a result for hypercubes due to Czygrinow and Wagner.

• N. Bushaw, C. E. Larson and N. Van Cleemput. Automated conjecturing for mathematics teaching and research projects. PRIMUS 35 (6) (2025), 636–653.
• N. Bushaw, V. Gupta, C. E. Larson, S. Loeb, M. Norge, J. Parrish, N. Van Cleemput, J. Yirka and G. Wu. Automated conjectures for graph Hamiltonicity. Involve 18 (1) (2025), 79–89.
  Abstract

  We present results on new sufficient or necessary conditions for the existence of a Hamilton cycle in a graph. We are especially interested in finding conditions which are not implied by any of a number of well-known theorems in the literature on graph hamiltonicity. We also report a number of unresolved conjectures.

• N. Bushaw and G. Hurlbert. Thresholds for zero-sums with small cross numbers in abelian groups. INTEGERS 24 (2024), #A93, 12 pp. INTEGERS · arXiv
  Abstract

  For an additive group $\mathrm{\Gamma }$ the sequence $S=(g_1,\dots ,g_t)$ of elements of $\mathrm{\Gamma }$ is a zero-sum sequence if $g_1+\cdots +g_t=0_{\mathrm{\Gamma }}$. The cross number of $S$ is defined to be the sum ${\sum }_{i=1}^{k}1\mathrm{/}\mathrm{\mid }{g}_i\mathrm{\mid }$, where $\mathrm{\mid }{g}_i\mathrm{\mid }$ denotes the order of $g_i$ in $\mathrm{\Gamma }$. Call $S$ good if it contains a zero-sum subsequence with cross number at most 1. In 1993, Geroldinger proved that if $\mathrm{\Gamma }$ is abelian then every length $\mathrm{\mid }\mathrm{\Gamma }\mathrm{\mid }$ sequence of its elements is good, generalizing a 1989 result of Lemke and Kleitman that had proved an earlier conjecture of Erdős and Lemke. In 1989 Chung re-proved the Lemke and Kleitman result by applying a theorem of graph pebbling, and in 2005, Elledge and Hurlbert used graph pebbling to re-prove and generalize Geroldinger's result. Here we use probabilistic theorems from graph pebbling to derive a threshold version of Geroldinger's theorem for abelian groups of a certain form. Specifically, we prove that if $p_1,\dots ,p_d$ are (not necessarily distinct) primes and ${\mathrm{\Gamma }}_k$ has the form ${\prod }_{i=1}^d{\mathbb{Z}}_{p_i^k}$ then there is a function $\tau =\tau (k)$ (which we specify in Theorem 4) with the following property: if $t-\tau \to \mathrm{\infty }$ as $k\to \mathrm{\infty }$ then the probability that $S$ is good in ${\mathrm{\Gamma }}_k$ tends to 1.

• N. Bushaw, B. Cody, L. Freeman and T. Whitaker. The music and mathematics of maximal evenness in graphs. Bridges 2024 Conference Proceedings, 61–68. Bridges archive (PDF) · arXiv
  Abstract

  We use the concept of electric potential energy from physics, the mathematical field of graph theory, and the notion of majorization to study maximal evenness in a broader mathematical context than what was previously possible, so that we can go beyond the well-known one-dimensional maximally even sets into higher dimensional and more geometrically complex territory. We investigate musical connections between certain generalizations of maximally even sets, one of the oldest Puerto Rican musical traditions of African origin called bomba, and with certain scales ranging from the familiar to the esoteric.

• N. Bushaw, B. Conka, V. Gupta, A. Kierans, H. LaFayette, C. E. Larson, K. McCall, A. Mulyar, C. Sullivan, S. Taylor, E. Wainright, E. Wilson, G. Wu and S. Loeb. Bootstrap percolation via automated conjecturing. Ars Mathematica Contemporanea 23 (3) (2023).
  Abstract

  Bootstrap percolation is a simple monotone cellular automaton with a long history in physics, computer science, and discrete mathematics. In k-neighbor bootstrap percolation, a collection of vertices are initially infected. Vertices with at least k infected neighbors subsequently become infected; the process continues until no new vertices become infected. In this paper, we hunt for graphs which can become entirely infected from initial sets which are as small as possible. We use automated conjecture-generating software and a large group lab-based model as a fundamental part of our exploration.

• N. Bushaw, D. Johnston and P. Rombach. Rainbow saturation. Graphs and Combinatorics 38 (5) (2022), article 166. arXiv
  Abstract

  We introduce a notion of rainbow saturation and the corresponding rainbow saturation number. This is the saturation version of the rainbow Turán numbers whose systematic study was initiated by Keevash, Mubayi, Sudakov, and Verstraëte. We give examples of graphs for which the rainbow saturation number is bounded away from the ordinary saturation number. This includes all complete graphs $K_n$ for $n\ge 4$, and several bipartite graphs. It is notable that there are non-bipartite graphs for which this is the case, as this does not happen when it comes to the rainbow extremal number versus the traditional extremal number. We also show that saturation numbers are linear for a large class of graphs, providing a partial rainbow analogue of a well known theorem of Kásonyi and Tuza. We conclude this paper with related open questions and conjectures.

• N. Bushaw and N. Kettle. Forbidding multiple copies of forestable graphs. Graphs and Combinatorics 36 (3) (2020), 459–467.
  Abstract

  The Turán number of a graph H is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. We call a graph H forestable if it is cyclic, bipartite, and contains a vertex v such that $H[V\setminus v]$ is a forest. For a forestable graph H, we determine $\text{ex}(n,k\cdot H)$ exactly as a function of $\text{ex}(n,H)$. This is related to earlier work of the authors on the Turán numbers for equibipartite forests.

• N. Bushaw and D. W. Cranston. A note on bootstrap percolation thresholds in plane tilings using regular polygons. Australasian Journal of Combinatorics 74 (3) (2019), 486–497. Australasian J. Combinatorics (PDF) · arXiv
  Abstract

  In $k$-bootstrap percolation, we fix $p\in (0,1)$, an integer $k$, and a plane graph $G$. Initially, we infect each face of $G$ independently with probability $p$. Infected faces remain infected forever, and if a healthy (uninfected) face has at least $k$ infected neighbors, then it becomes infected. For fixed $G$ and $p$, the percolation threshold is the largest $k$ such that eventually all faces become infected, with probability at least $1\mathrm{/}2$. For a large class of infinite graphs, we show that this threshold is independent of $p$.

  We consider bootstrap percolation in tilings of the plane by regular polygons. A vertex type in such a tiling is the cyclic order of the faces that meet a common vertex. First, we determine the percolation threshold for each of the Archimedean lattices. More generally, let $\mathcal{T}$ denote the set of plane tilings $T$ by regular polygons such that if $T$ contains one instance of a vertex type, then $T$ contains infinitely many instances of that type. We show that no tiling in $\mathcal{T}$ has threshold 4 or more. Further, the only tilings in $\mathcal{T}$ with threshold 3 are four of the Archimedean lattices. Finally, we describe a large subclass of $\mathcal{T}$ with threshold 2.

• N. Bushaw, C. E. Larson and N. Van Cleemput. Automated conjecturing VII: the graph brain project & big mathematics. Preprint. arXiv
  Abstract

  The Graph Brain Project is an experiment in how the use of automated mathematical discovery software, databases, large collaboration, and systematic investigation provide a model for how mathematical research might proceed in the future.

  Our Project began with the development of a program that can be used to generate invariant-relation and property-relation conjectures in many areas of mathematics. This program can produce conjectures which are not implied by existing (published) theorems. Here we propose a new approach to push forward existing mathematical research goals---using automated mathematical discovery software. We suggest how to initiate and harness large-scale collaborative mathematics. We envision mathematical research labs similar to what exist in other sciences, new avenues for funding, new opportunities for training students, and a more efficient and effective use of published mathematical research.

  And our experiment in graph theory can be imitated in many other areas of mathematical research. Big Mathematics is the idea of large, systematic, collaborative research on problems of existing mathematical interest. What is possible when we put our skills, tools, and results together systematically?

• N. Bushaw, A. Czygrinow and J. Yie. Even cycles in dense graphs. Preprint. arXiv
  Abstract

  We will show that for $\alpha >0$ there is $n_0$ such that if $G$ is a graph on $n\ge n_0$ vertices such that $\alpha n<\delta (G)<(n-1)\mathrm{/}2$, then for every $n_1+n_2+\cdots +n_l=\delta (G)$, $G$ contains a disjoint union of $C_{2n_1},C_{2n_2},\dots ,C_{2n_l}$ unless $G$ has a very specific structure.

• J. Balogh, N. Bushaw, M. Collares, H. Liu, R. Morris and M. Sharifzadeh. The typical structure of graphs with no large cliques. Combinatorica 37 (4) (2017), 617–632. arXiv
  Abstract

  In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed $r\in \mathbb{N}$, almost every $n$-vertex $K_{r+1}$-free graph is $r$-partite. In this paper we extend this result to all functions $r=r(n)$ with $r⩽(\log n{)}^{1\mathrm{/}4}$. The proof combines a new (close to sharp) supersaturation version of the Erdős-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollobás and Simonovits.

• N. Bushaw, K. Gunderson and S. Kalikow. Random-step Markov processes. Israel Journal of Mathematics 216 (1) (2016), 181–214. arXiv
  Abstract

  We explore two notions of stationary processes. The first is called a random-step Markov process in which the stationary process of states, $(X_i{)}_{i\in \mathbb{Z}}$ has a stationary coupling with an independent process on the positive integers, $(L_i{)}_{i\in \mathbb{Z}}$ of `random look-back distances'. That is,

  $L_0$ is independent of the `past states', $(X_i,L_i{)}_{i<0}$, and for every positive integer $n$, the probability distribution on the `present', $X_0$, conditioned on the event $\{L_0=n\}$ and on the past is the same as the probability distribution on $X_0$ conditioned on the `$n$-past', $(X_i{)}_{-n\le i<0}$ and $\{L_0=n\}$. A random Markov process is a generalization of a Markov chain of order $n$ and has the property that the distribution on the present given the past can be uniformly approximated given the $n$-past, for $n$ sufficiently large. Processes with the latter property are called uniform martingales, closely related to the notion of a `continuous $g$-function'.

  We show that every stationary process on a countable alphabet that is a uniform martingale and is dominated by a finite measure is also a random Markov process and that the random variables $(L_i{)}_{i\in \mathbb{Z}}$ and associated coupling can be chosen so that the distribution on the present given the $n$-past and the event $\{L_0=n\}$ is `deterministic': all probabilities are in $\{0,1\}$. In the case of finite alphabets, those random-step Markov processes for which $L_0$ can be chosen with finite expected value are characterized. For stationary processes on an uncountable alphabet, a stronger condition is also considered which is sufficient to imply that a process is a random Markov processes. In addition, a number of examples are given throughout to show the sharpness of the results.

• N. Bushaw, M. Collares, R. Morris and P. Smith. The sharp threshold for maximum-size sum-free subsets in even-order abelian groups. Combinatorics, Probability and Computing 24 (4) (2015), 609–640. arXiv
  Abstract

  We study sum-free sets in sparse random subsets of even order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group. This theorem extends recent work of Balogh, Morris and Samotij, who resolved the case G = Z_{2n}, and who obtained a weaker threshold (up to a constant factor) in general.

• N. Bushaw and N. Kettle. Turán numbers for forests of paths in hypergraphs. SIAM Journal on Discrete Mathematics 28 (2) (2014), 711–721. arXiv
  Abstract

  The Turán number of an r-uniform hypergraph H is the maximum number of edges in any r-graph on n vertices which does not contain H as a subgraph. Let P_l^(r) denote the family of r-uniform loose paths on l edges, F(k,l) denote the family of hypergraphs consisting of k disjoint paths from P_l^(r), and P'_l^(r) denote an r-uniform linear path on l edges. We determine precisely ex_r(n;F(k,l)) and ex_r(n;k*P'_l^(r)), as well as the Turán numbers for forests of paths of differing lengths (whether these paths are loose or linear) when n is appropriately large dependent on k,l,r, for r>=3. Our results build on recent results of Füredi, Jiang, and Seiver who determined the extremal numbers for individual paths, and provide more hypergraphs whose Turan numbers are exactly determined.

• N. Bushaw. Problems in extremal combinatorics. PhD dissertation, University of Memphis, May 2012. Advisor: Béla Bollobás. Memphis repository
  Abstract

  This dissertation is divided into two major sections. Chapters 1 to 4 are concerned with Turán type problems for disconnected graphs and hypergraphs. In Chapter 5, we discuss an unrelated problem dealing with the equivalence of two notions of stationary processes. The Turán number of a graph H, ex(n,H), is the maximum number of edges in any n-vertex graph which is H-free. We discuss the history and results in this area, focusing particularly on the degenerate case for bipartite graphs.

  Let Pl denote a path on l vertices, and k*Pl denote k vertex-disjoint copies of Pl. We determine ex(n,k*P3) for n appropriately large, confirming a conjecture of Gorgol. Further, we determine ex(n,k*Pl) for arbitrary l, and n appropriately large. We provide background on the famous Erdös-Sós conjecture, and conditional on its truth we determine ex(n,H) when H is an equibipartite forest, for appropriately large n. In Chapter 4, we prove similar results in hypergraphs.

  We first discuss the related results for extremal numbers of hyperpaths, before proving the extremal numbers for multiple copies of a loose path of fixed length, and the corresponding result for linear paths. We extend this result to forests of loose hyperpaths, and linear hyperpaths. We note here that our results for loose paths, while tight, do not give the extremal numbers in their classical form; much more detail on this is given in Chapter 4.

  InChapter 5, we discuss two notions of stationary processes. Roughly, a process is a uniform martingale if it can be approximated arbitrarily well by a process in which the letter distribution depends only on a finite amount of the past. A random Markov process is a process with a coupled `look back' time; that is, to determine the letter distribution, it suffices to choose a random look-back time, and then the distribution depends only on the past up to this time.

  Kalikow proved that on a binary alphabet, any uniform martingale is also a random Markov process. We extend this result to any finite alphabet.

• N. Bushaw, P. Csorba, L. Erickson, D. Gerbner, D. Piguet, A. Riet, T. Terpai and D. K. Vu. Large matchings with few colors. Preprint. arXiv
  Abstract

  Let $K_n^r$ denote the complete $r$-uniform hypergraph on $n$ vertices. A matching $M$ in a hypergraph is a set of pairwise vertex disjoint edges. Recent Ramsey-type results rely on lemmas about the size of monochromatic matchings. A starting point for this study comes from a well-known result of Alon, Frankl, and Lovász (1986). Our motivation is to find the smallest $n$ such that every $t$-coloring of $K_n^r$ contains an $s$-colored matching of size $k$. It has been conjectured that in every coloring of the edges of $K_n^r$ with 3 colors there is a 2-colored matching of size at least $k$ provided that $n\ge kr+\lfloor \frac{k-1}{r+1}\rfloor$. The smallest test case is when $r=3$ and $k=4$. We prove that in every 3-coloring of the edges of $K_{12}^3$ there is a 2-colored matching of size 4.

• N. Bushaw and N. Kettle. Turán numbers of multiple paths and equibipartite forests. Combinatorics, Probability and Computing 20 (6) (2011), 837–853. arXiv
  Abstract

  The Turán number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P_l denote a path on l vertices, and kP_l denote k vertex-disjoint copies of P_l. We determine ex(n, kP_3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex (n, kP_l) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erdős-Sós conjecture, and conditional on its truth we determine ex(n;H) when H is an equibipartite forest, for appropriately large n.

Talks

Teaching

Fall 2026

• MATH 300 — Introduction to Mathematical Reasoning
  Section 004 · CRN 39277
• MATH 350 — Introductory Combinatorics
  Section 001 · CRN 44237

I work from the axioms of Federico Ardila-Mantilla, as set out in Todos Cuentan (PDF), Notices of the AMS 63 (2016), 1164–1170.

1. Axiom 1. Mathematical talent is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
2. Axiom 2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
3. Axiom 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
4. Axiom 4. Every student deserves to be treated with dignity and respect.

Who?

I'm an Associate Professor in the Department of Mathematics and Statistics at Virginia Commonwealth University.

Before that, I was a Visiting Assistant Professor at Arizona State University, and a research postdoc at Instituto Nacional de Matemática Pura e Aplicada in Rio de Janeiro. I completed my doctorate under the supervision of Prof. Béla Bollobás at the University of Memphis in May 2012; this followed an M.S. at Western Washington University under the supervision of Dr. Amites Sarkar, and a B.A. at the University of Colorado. I am greatly indebted to all my teachers and to those around me who have helped me throughout my academic career, although I attempt no comprehensive list here. The advisor graph is the closest thing to one.

My academic interests lie primarily in the mysteries and details of mathematics, particularly extremal and probabilistic combinatorics. Outside academia, I enjoy weird jazz, loud punk rock, Fender guitars and Martin guitars, skiing, snowboarding, as well as life, the universe, and everything.

• VCU My official department profile.
• ORCID 0000-0003-2441-5713
• Profiles Google Scholar · arXiv · dblp · MathSciNet · genealogy project
• Code github.com/thenealon, including the source of this site.
• Family Donald “Beh” Bushaw — my illustrious grandfather, on MathSciNet.
• Pledges Theoretical Computer Scientists for Future
• On AI The Leiden Declaration on Artificial Intelligence and Mathematics, endorsed by the International Mathematical Union.

Extras

Juried objects

• Cyclic and Möbius Pianos (2026)
  Bridges 2026 Exhibition of Mathematical Art, Craft, and Design · University of Galway, Ireland

  3D-printed keyboards laid out on cycles and Möbius ladders.

• Alternative Pianos (2024)
  Bridges 2024 Exhibition of Mathematical Art, Craft, and Design · Richmond VA

  3D-printed keyboards built on cycles and Möbius ladders.