Diophantine approximation and coloring

We demonstrate how connections between graph theory and Diophantine approximation can be used in conjunction to give simple and accessible proofs of seemingly difficult results in both subjects.Comment: 16 pages, pre-publication version of paper which will appear in American Mathematical Monthl

Separating topological recurrence from measurable recurrence: exposition and extension of Kriz's example

We prove that for every infinite set $E\subset \mathbb Z$, there is a set $S\subset E-E$ which is a set of topological recurrence and not a set of measurable recurrence. This extends a result of Igor Kriz, proving that there is a set of topological recurrence which is not a set of measurable recurrence. Our construction follows Kriz's closely, and this paper can be considered an exposition of the original argument.Comment: 20 Pages; v.2 incorporates referee suggestion

Recursive Linear Bounds for the Vertex Chromatic Number of the Pancake Graph

The pancake graph has been the subject of research. While studies on the various aspects of the graph are abundant, results on the chromatic properties may be further enhanced. Revolving around such context, the paper advances an alternative method to produce novel linear bounds for the vertex chromatic number of the pancake graph. The accompanying demonstration takes advantage of symmetries inherent to the graph, capturing the prefix reversal of subsequences through a homomorphism. Contained within the argument is the incorporation of known vertex chromatic numbers for certain orders of pancake graphs, rendering tighter bounds possible upon the release of new findings. In closing, a comparison with existing bounds is done to establish the relative advantage of the proposed technique

On the Spectrum of the Derangement Graph

We derive several interesting formulae for the eigenvalues of the derangement graph and use them to settle affirmatively a conjecture of Ku regarding the least eigenvalue

Diophantine approximation and coloring

We demonstrate how connections between graph theory and Diophantine approximation can be used in conjunction to give simple and accessible proofs of seemingly difficult results in both subjects

Special cases and equivalent forms of Katznelson's problem on recurrence

We make three observations regarding a question popularized by Katznelson: is every subset of $\mathbb Z$ which is a set of Bohr recurrence is also a set of topological recurrence? (i) If $G$ is a countable abelian group and $E\subset G$ is an $I_0$ set, then every subset of $E-E$ which is a set of Bohr recurrence is also a set of topological recurrence. In particular every subset of $\{2^n-2^m : n,m\in \mathbb N\}$ which is a set of Bohr recurrence is a set of topological recurrence. (ii) Let $\mathbb Z^{\omega}$ be the direct sum of countably many copies of $\mathbb Z$ with standard basis $E$. If every subset of $(E-E)-(E-E)$ which is a set of Bohr recurrence is also a set of topological recurrence, then every subset of every countable abelian group which is a set of Bohr recurrence is also a set of topological recurrence. (iii) Fix a prime $p$ and let $\mathbb F_p^\omega$ be the direct sum of countably many copies of $\mathbb Z/p\mathbb Z$ with basis $(\mathbf e_i)_{i\in \mathbb N}$. If for every $p$-uniform hypergraph with vertex set $\mathbb N$ and edge set $\mathcal F$ having infinite chromatic number, the Cayley graph on $\mathbb F_p^\omega$ determined by $\{\sum_{i\in F}\mathbf e_i:F\in \mathcal F\}$ has infinite chromatic number, then every subset of $\mathbb F_p^\omega$ which is a set of Bohr recurrence is a set of topological recurrence.Comment: 20 pages; v.2 incorporates referee suggestion

Pseudo-random graphs

Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in proving an enormous number of combinatorial statements, making their role quite hard to overestimate. Their tremendous success serves as a natural motivation for the following very general and deep informal questions: what are the essential properties of random graphs? How can one tell when a given graph behaves like a random graph? How to create deterministically graphs that look random-like? This leads us to a concept of pseudo-random graphs and the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page

Two Erdos problems on lacunary sequences: Chromatic number and Diophantine approximation

Let ${n_k}$ be an increasing lacunary sequence, i.e., $n_{k+1}/n_k>1+r$ for some $r>0$. In 1987, P. Erdos asked for the chromatic number of a graph $G$ on the integers, where two integers $a,b$ are connected by an edge iff their difference $|a-b|$ is in the sequence ${n_k}$. Y. Katznelson found a connection to a Diophantine approximation problem (also due to Erdos): the existence of $x$ in $(0,1)$ such that all the multiples $n_j x$ are at least distance $\delta(x)>0$ from the set of integers. Katznelson bounded the chromatic number of $G$ by $Cr^{-2}|\log r|$. We apply the Lov\'asz local lemma to establish that $\delta(x)>cr|\log r|^{-1}$ for some $x$, which implies that the chromatic number of $G$ is at most $Cr^{-1} |\log r|$. This is sharp up to the logarithmic factor.Comment: 9 page

Variations on topological recurrence

Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate finite versions of recurrence, and describe connections to combinatorial problems. In particular, we show that sets of Bohr recurrence (meaning sets of recurrence for rotations) suffice for recurrence in nilsystems. Additionally, we prove an extension of this property for multiple recurrence in affine systems