Related papers: Lower bounds on geometric Ramsey functions

For graphs $F$ and $H$, we say $F$ is Ramsey for $H$ if every $2$-coloring of the edges of $F$ contains a monochromatic copy of $H$. The graph $F$ is Ramsey $H$-minimal if $F$ is Ramsey for $H$ and there is no proper subgraph $F'$ of $F$ so…

In this paper, we determine the almost sure values of the $\Phi$-dimensions of random measures $\mu$ supported on random Moran sets in $\R^d$ that satisfy a uniform separation condition. This paper generalizes earlier work done on random…

A graph is $(t_1, t_2)$-Ramsey if any red-blue coloring of its edges contains either a red copy of $K_{t_1}$ or a blue copy of $K_{t_2}$. The size Ramsey number is the minimum number of edges contained in a $(t_1,t_2)$-Ramsey graph.…

In the dual $L_{\Phi^*}$ of a $\Delta_2$-Orlicz space $L_\Phi$, that we call a dual Orlicz space, we show that a proper (resp. finite) convex function is lower semicontinuous (resp. continuous) for the Mackey topology…

Resolving a problem of Conlon, Fox, and R\"{o}dl, we construct a family of hypergraphs with arbitrarily large tower height separation between their $2$-colour and $q$-colour Ramsey numbers. The main lemma underlying this construction is a…

Let $I$ be a homogeneous ideal in the polynomial ring $R = k[z_1, \cdots, z_n]$ , where $k$ is an algebraically closed field of characteristic zero. Macaulay's Theorem provides constraints on the Hilbert function of $I$ or $R/I$ from one…

For any positive integer $k$ and nonnegative integer $m$, we consider the symmetric function $G\left( k,m\right)$ defined as the sum of all monomials of degree $m$ that involve only exponents smaller than $k$. We call $G\left( k,m\right)$ a…

The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemer{\'e}di regularity lemma in graph theory. It shows that for any abelian group $G$ and any bounded function $f:G \to [0,1]$, there exists a subgroup…

This paper is concerned with a class of nonmonotone descent methods for minimizing a proper lower semicontinuous KL function $\Phi$, which generates a sequence satisfying a nonmonotone decrease condition and a relative error tolerance.…

A symmetric subset of the reals is one that remains invariant under some reflection x --> c-x. Given 0 < x < 1, there exists a real number D(x) with the following property: if 0 < d < D(x), then every subset of [0,1] with measure x contains…

Let $\Omega\subset{\mathbb R}^n$ be a relatively compact domain. A finite collection of real-valued functions on $\Omega$ is called a \emph{Noetherian chain} if the partial derivatives of each function are expressible as polynomials in the…

Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f: P\rightarrow P'$ and for any $X, Y\in P$, $X\leq _P Y$ if and only of $f(X)\leq_{P'}…

A matrix $\Phi \in \mathbb{R}^{Q \times N}$ satisfies the restricted isometry property if $\|\Phi x\|_2^2$ is approximately equal to $\|x\|_2^2$ for all $k$-sparse vectors $x$. We give a construction of RIP matrices with the optimal $Q =…

Given posets $\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k$, let the {\em Boolean Ramsey number} $R(\mathbf{P}_1,\mathbf{P}_2,\ldots,\mathbf{P}_k)$ be the minimum number $n$ such that no matter how we color the elements in the Boolean…

For a field $\mathbb{F}$ and integers $d, k$ and $\ell$, a set $A \subseteq \mathbb{F}^d$ is called $(k,\ell)$-nearly orthogonal if all vectors in $A$ are non-self-orthogonal and every $k+1$ vectors in $A$ contain $\ell + 1$ pairwise…

We consider fast, provably accurate algorithms for approximating functions on the $d$-dimensional torus, $f: \mathbb{ T }^d \rightarrow \mathbb{C}$, that are sparse (or compressible) in the Fourier basis. In particular, suppose that the…

In this thesis, we consider semi-algebraic sets over a real closed field $R$ defined by quadratic polynomials. Semi-algebraic sets of $R^k$ are defined as the smallest family of sets in $R^k$ that contains the algebraic sets as well as the…

The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in…

We prove a structural result for degree-$d$ polynomials. In particular, we show that any degree-$d$ polynomial, $p$ can be approximated by another polynomial, $p_0$, which can be decomposed as some function of polynomials $q_1,...,q_m$ with…

The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is one…