High-Girth Matrices and Polarization

The girth of a matrix is the least number of linearly dependent columns, in contrast to the rank which is the largest number of linearly independent columns. This paper considers the construction of {\it high-girth} matrices, whose probabilistic girth is close to its rank. Random matrices can be used to show the existence of high-girth matrices with constant relative rank, but the construction is non-explicit. This paper uses a polar-like construction to obtain a deterministic and efficient construction of high-girth matrices for arbitrary fields and relative ranks. Applications to coding and sparse recovery are discussed

Polynomials that vanish to high order on most of the hypercube

Motivated by higher vanishing multiplicity generalizations of Alon's Combinatorial Nullstellensatz and its applications, we study the following problem: for fixed $k\geq 1$ and $n$ large with respect to $k$, what is the minimum possible degree of a polynomial $P\in \mathbb{R}[x_1,\dots,x_n]$ with $P(0,\dots,0)\neq 0$ such that $P$ has zeroes of multiplicity at least $k$ at all points in $\{0,1\}^n\setminus \{(0,\dots,0)\}$? For $k=1$, a classical theorem of Alon and F\"uredi states that the minimum possible degree of such a polynomial equals $n$. In this paper, we solve the problem for all $k\geq 2$, proving that the answer is $n+2k-3$. As an application, we improve a result of Clifton and Huang on configurations of hyperplanes in $\mathbb{R}^n$ such that each point in $\{0,1\}^n\setminus \{(0,\dots,0)\}$ is covered by at least $k$ hyperplanes, but the point $(0,\dots,0)$ is uncovered. Surprisingly, the proof of our result involves Catalan numbers and arguments from enumerative combinatorics.Comment: 19 page

Ramsey multiplicity and the Tur\'an coloring

Extending an earlier conjecture of Erd\H{o}s, Burr and Rosta conjectured that among all two-colorings of the edges of a complete graph, the uniformly random coloring asymptotically minimizes the number of monochromatic copies of any fixed graph $H$. This conjecture was disproved independently by Sidorenko and Thomason. The first author later found quantitatively stronger counterexamples, using the Tur\'an coloring, in which one of the two colors spans a balanced complete multipartite graph. We prove that the Tur\'an coloring is extremal for an infinite family of graphs, and that it is the unique extremal coloring. This yields the first determination of the Ramsey multiplicity constant of a graph for which the Burr--Rosta conjecture fails. We also prove an analogous three-color result. In this case, our result is conditional on a certain natural conjecture on the behavior of two-color Ramsey numbers.Comment: 37 page

On linear-algebraic notions of expansion

A fundamental fact about bounded-degree graph expanders is that three notions of expansion -- vertex expansion, edge expansion, and spectral expansion -- are all equivalent. In this paper, we study to what extent such a statement is true for linear-algebraic notions of expansion. There are two well-studied notions of linear-algebraic expansion, namely dimension expansion (defined in analogy to graph vertex expansion) and quantum expansion (defined in analogy to graph spectral expansion). Lubotzky and Zelmanov proved that the latter implies the former. We prove that the converse is false: there are dimension expanders which are not quantum expanders. Moreover, this asymmetry is explained by the fact that there are two distinct linear-algebraic analogues of graph edge expansion. The first of these is quantum edge expansion, which was introduced by Hastings, and which he proved to be equivalent to quantum expansion. We introduce a new notion, termed dimension edge expansion, which we prove is equivalent to dimension expansion and which is implied by quantum edge expansion. Thus, the separation above is implied by a finer one: dimension edge expansion is strictly weaker than quantum edge expansion. This new notion also leads to a new, more modular proof of the Lubotzky--Zelmanov result that quantum expanders are dimension expanders.Comment: 23 pages, 1 figur

Ramsey goodness of books revisited

The Ramsey number $r(G,H)$ is the minimum $N$ such that every graph on $N$ vertices contains $G$ as a subgraph or its complement contains $H$ as a subgraph. For integers $n \geq k \geq 1$, the $k$-book $B_{k,n}$ is the graph on $n$ vertices consisting of a copy of $K_k$, called the spine, as well as $n-k$ additional vertices each adjacent to every vertex of the spine and non-adjacent to each other. A connected graph $H$ on $n$ vertices is called $p$-good if $r(K_p,H)=(p-1)(n-1)+1$. Nikiforov and Rousseau proved that if $n$ is sufficiently large in terms of $p$ and $k$, then $B_{k,n}$ is $p$-good. Their proof uses Szemer\'edi's regularity lemma and gives a tower-type bound on $n$. We give a short new proof that avoids using the regularity method and shows that every $B_{k,n}$ with $n \geq 2^{k^{10p}}$ is $p$-good. Using Szemer\'edi's regularity lemma, Nikiforov and Rousseau also proved much more general goodness-type results, proving a tight bound on $r(G,H)$ for several families of sparse graphs $G$ and $H$ as long as $|V(G)| < \delta |V(H)|$ for a small constant $\delta > 0$. Using our techniques, we prove a new result of this type, showing that $r(G,H) = (p-1)(n-1)+1$ when $H =B_{k,n}$ and $G$ is a complete $p$-partite graph whose first $p-1$ parts have constant size and whose last part has size $\delta n$, for some small constant $\delta>0$. Again, our proof does not use the regularity method, and thus yields double-exponential bounds on $\delta$.Comment: 21 page

A short proof of the canonical polynomial van der Waerden theorem

We present a short new proof of the canonical polynomial van der Waerden theorem, recently established by Girao [arXiv:2004.07766].Comment: 2 page