On Kuroda's proof of Hilbert's fourteenth problem in dimensions three and four

In this expository paper we present short simple proofs of Conway-Gordon-Sachs' theorem on intrinsic linking in three-dimensional space, as well as van Kampen-Flores' and Ummel's theorems on intrinsic intersections. The latter are related to nonrealizability of certain hypergraphs in four-dimensional space. The proofs use a reduction to lower dimensions which allows to exhibit relation between these results. We use elementary language which allows to present the main ideas wi...

We give a short and relatively elementary proof of the Hilton-Milner Theorem.

There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a simple smooth closed curve in the plane has at least four extreme values. This theorem admits a natural discretization to Euclidean polygons due to O. Musin. In this article we adapt the techniques of Musin and prove a discrete four vertex th...

This note points out a gap in the proof of the main theorem of the article "Birationally rigid hypersurfaces" published in Invent. Math. 192 (2013), 533-566, and provides a new proof of the theorem.

The general structure of the paper should be remaid. Hence author removed this paper from arXiv.

Borsuk asked in 1933 if every set of diameter 1 in $R^d$ can be covered by $d+1$ sets of smaller diameter. In 1993, a negative solution, based on a theorem by Frankl and Wilson, was given by Kahn and Kalai. In this paper I will present questions related to Borsuk's problem.

In the paper based on the question of Zhang and L\"{u}[15], we present one theorem which will improve and extend the results of Banerjee-Majumder [2] and a recent result of Li-Huang [9].

This work is a continuation of [1]. As in the previous article, here we will describe some interesting ideas and a lot of new theorems in plane geometry related to them.

We present a collection of research questions on cubic surfaces in 3-space. These questions inspired a collection of papers to be published in a special issue of the journal Le Matematiche. This article serves as the introduction to that issue. The number of questions is meant to match the number of lines on a cubic surface. We end with a list of problems that are open.

This article contains a short and entertaining list of unsolved problems in Plane Geometry. Their statement may seem naive and can be understood at an elementary level. But their solutions have refused to appear for forty years in the best case.