Related papers: On Two of Erd\"os's Open Problems

We expand upon some topics reviewed and sketched in a book to appear with more details, embellishments, and some new material of a speculative nature.

Some mathematical properties of the Erd\"os number and its formal equivalents are shown. An informetric equivalent is presented: the Rousseau number. This contribution honors Ronald Rousseau for his 73th birthday.

In this paper, we investigate two variations on the so-called persistence problem of Sloane: the shifted version, which was introduced by Wagstaff; and the nonzero version, proposed by Erd\H{o}s. We explore connections between these…

In this short paper we review and extract some features of the Fredholm Alternative problem .

This is a brief review of the present status, of some recent developments and of the open challenges in string/M theory.

Erd\H{o}s proved that there are graphs with arbitrarily large girth and chromatic number. We study the extension of this for generalized chromatic numbers.

This is an appendix to our paper "An update of the Hirsch Conjecture" (arXiv:0907.1186), containing proofs of some of the results and comments that were omitted in it.

In this short note, we answer two questions of Chen and Ruzsa negatively and answer a problem of Ma and Chen affirmatively.

In this very short note, we give a counterexample to a recent conjecture of Gilmer which would have implied the union-closed conjecture.

I settle a conjecture of Andrews related to the Alladi-Schur polynomials. In addition, I give further relations and implications to two families of polynomials related to the Alladi-Schur polynomials.

We establish some new generalizations of Erd\H{o}s-Mordell inequality by adding weights to its terms. Using these generalizations, we derived strengthened versions of the original Erd\H{o}s-Mordell inequality. We also found two other…

We explore variants of Erd\H os' unit distance problem concerning dot products between successive pairs of points chosen from a large finite subset of either $\mathbb F_q^d$ or $\mathbb Z_q^d,$ where $q$ is a power of an odd prime.…

This note is about variations on a theorem of Bers about short pants decompositions of surfaces. It contains a version for surfaces with boundary but also a slight improvement on the best known bound for closed surfaces.

We prove that certain problems naturally arising in knot theory are NP--hard or NP--complete. These are the problems of obtaining one diagram from another one of a link in a bounded number of Reidemeister moves, determining whether a link…

Erd\H{o}s asked many mathematical questions. Some lead to exciting research, others turned out to be easily solved. In this article, we provide evidence that one of his questions, Erd\H{o}s problem \#278 , has no general answer. We do so by…

We briefly review superstring theories, highlighting the important concepts, developments, and open problems of the subject.

Open-string theories may be related to suitable models of oriented closed strings. The resulting construction of ``open descendants'' is illustrated in a few simple cases that exhibit some of its key features.

We present a brief introduction to the theory of operator limits of random matrices to non-experts. Several open problems and conjectures are given. Connections to statistics, integrable systems, orthogonal polynomials, and more, are…

This short note contains elementary evaluations of some Euler sums.

Several open questions are discussed. The topics include cohomology of current and related Lie algebras, algebras represented as the sum of subalgebras, structures and phenomena peculiar to characteristic $2$, and variations on themes of…