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The log-density method is a powerful algorithmic framework which in recent years has given rise to the best-known approximations for a variety of problems, including Densest-$k$-Subgraph and Bipartite Small Set Vertex Expansion. These…
For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is guaranteed…
The Fermi-Pasta-Ulam $\alpha$-model of harmonic oscillators with cubic anharmonic interactions is studied from a statistical mechanical point of view. Systems of N= 32 to 128 oscillators appear to be large enough to suggest statistical…
We study a model of spinless fermions with infinite nearest-neighbor repulsion on the square ladder which has microscopic supersymmetry. It has been conjectured that in the continuum the model is described by the superconformal minimal…
The problem of bounding the size of a set system under various intersection restrictions has a central place in extremal combinatorics. We investigate the maximum number of disjoint pairs a set system can have in this setting. In…
We investigate dephasing in open quantum chaotic systems in the limit of large system size to Fermi wavelength ratio, $L/\lambda_F >> 1$. We semiclassically calculate the weak localization correction $g^{wl}$ to the conductance for a…
Let $\mathcal{Q}$ be a vertex subset problem on graphs. In a reconfiguration variant of $\mathcal{Q}$ we are given a graph $G$ and two feasible solutions $S_s, S_t\subseteq V(G)$ of $\mathcal{Q}$ with $|S_s|=|S_t|=k$. The problem is to…
The rainbow Ramsey theorem states that every coloring of tuples where each color is used a bounded number of times has an infinite subdomain on which no color appears twice. The restriction of the statement to colorings over pairs (RRT22)…
Let n \geq l \geq 2 and q \geq 2. We consider the minimum N such that whenever we have N points in the plane in general position and the l-subsets of these points are colored with q colors, there is a subset S of n points all of whose…
Analogues of Ramsey's Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic…
A family of subsets $\mathcal{F}\subseteq {[n]\choose k}$ is called intersecting if any two of its members share a common element. Consider an intersecting family, a direct problem is to determine its maximal size and the inverse problem is…
A novel Thomas-Fermi (TF) approach to inhomogeneous superfluid Fermi-systems is presented and shown that it works well also in cases where the Local Density Approximation (LDA) breaks down. The novelty lies in the fact that the…
A set $X$ is called Euclidean Ramsey if, for any $k$ and sufficiently large $n$, every $k$-colouring of $\mathbb{R}^n$ contains a monochromatic congruent copy of $X$. This notion was introduced by Erd\H{o}s, Graham, Montgomery, Rothschild,…
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…
We address a supersaturation problem in the context of forbidden subposets. A family $\mathcal{F}$ of sets is said to contain the poset $P$ if there is an injection $i:P \rightarrow \mathcal{F}$ such that $p \le_P q$ implies $i(p) \subset i…
We investigate the intersection problem for finite semigroups, which asks for a given set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. We…
A family of vectors $A \subset [k]^n$ is said to be intersecting if any two elements of $A$ agree on at least one coordinate. We prove, for fixed $k \ge 3$, that the size of a symmetric intersecting subfamily of $[k]^n$ is $o(k^n)$, which…
A novel lattice approach is presented for studying systems comprising a large number of interacting nonrelativistic fermions. The construction is ideally suited for numerical study of fermions near unitarity--a strongly coupled regime…
Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every $d\geq 1$, if $P$ is a poset and the dimension of a subposet $B$ of $P$…
{\em Partial domination problem} is a generalization of the {\em minimum dominating set problem} on graphs. Here, instead of dominating all the nodes, one asks to dominate at least a fraction of the nodes of the given graph by choosing a…