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10 records were found.

## An analogue of Gromov's waist theorem for coloring the cube

It is proved that if we partition a $d$-dimensional cube into $n^d$ small cubes and color the small cubes into $m+1$ colors then there exists a monochromatic connected component consisting of at least $f(d, m) n^{d-m}$ small cubes.

## Extensions of theorems of Rattray and Makeev

We consider extensions of the Rattray theorem and two Makeev's theorems, showing that they hold for several maps, measures, or functions simultaneously, when we consider orthonormal $k$-frames in $\R^n$ instead of orthonormal basis (full frames). We also present new results on simultaneous partition of several measures into parts by $k$ mutually orthogonal hyperplanes. In the case $k=2$ we relate the Rattray and Makeev type results with the well known embedding problem for projective spaces.

## Estimating the higher symmetric topological complexity of spheres

Comment: This version has minor corrections compared to what published in AGT

## Kadets type theorems for partitions of a convex body

For convex partitions of a convex body $B$ we prove that we can put a homothetic copy of $B$ into each set of the partition so that the sum of homothety coefficients is $\ge 1$. In the plane the partition may be arbitrary, while in higher dimensions we need certain restrictions on the partition.

## Inscribing a regular octahedron into polytopes

We prove that any simple polytope (and some non-simple polytopes) in $\mathbb R^3$ admits an inscribed regular octahedron.

## Notes about the Caratheodory number

In this paper we give sufficient conditions for a compactum in $\mathbb R^n$ to have Carath\'{e}odory number less than $n+1$, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carath\'{e}odory theorem and give a Tverberg type theorem for families of convex compacta.

## Cutting the same fraction of several measures

Comment: 7 pages 2 figures

## Projective center point and Tverberg theorems

Comment: 10 pages

## Tur\'an numbers for $K_{s,t}$-free graphs: topological obstructions and algebraic constructions

Comment: Fixed a small mistake in the application of Proposition 1

## Borsuk-Ulam type theorems for metric spaces

In this paper we study the problems of the following kind: For a pair of topological spaces $X$ and $Y$ find sufficient conditions that under every continuous map $f : X\to Y$ a pair of sufficiently distant points is mapped to a single point.