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Andriy V. Bondarenko

Publications and source records attributed to Andriy V. Bondarenko.

6 recordsLinked to original sources
arXiv2022

Upper bounds on chromatic number of $\mathbb{E}^n$ in low dimensions

Let $χ(\mathbb{E}^n)$ denote the chromatic number of the Euclidean space $\mathbb{E}^n$, i.e., the smallest number of colors that can be used to color $\mathbb{E}^n$ so that no two points unit distance apart are of the same color. We present explicit constructions of colorings of $\mathbb{E}^n$ based on sublattice coloring schemes that establish the following new bounds: $χ(\mathbb{E}^5)\le 140$, $χ(\mathbb{E}^n)\le 7^{n/2}$ for $n\in\{6,8,24\}$, $χ(\mathbb{E}^7)\le 1372$, $χ(\mathbb{E}^{9})\leq 17253$, and $χ(\mathbb{E}^n)\le 3^n$ for all $n\le 38$ and $n=48,49$.

math.CO↗
arXiv2013

On Borsuk's conjecture for two-distance sets

In this paper we answer Larman's question on Borsuk's conjecture for two-distance sets. We find a two-distance set consisting of 416 points on the unit sphere in the dimension 65 which cannot be partitioned into 83 parts of smaller diameter. This also reduces the smallest dimension in which Borsuk's conjecture is known to be false. Other examples of two-distance sets with large Borsuk's numbers will be given.

math.MG↗
arXiv2012

On a family of strongly regular graphs with λ=1

In this paper, we give a complete description of strongly regular graphs with parameters ((n^2+3n-1)^2,n^2(n+3),1,n(n+1)). All possible such graphs are: the lattice graph $L_{3,3}$ with parameters (9,4,1,2), the Brouwer-Haemers graph with parameters (81,20,1,6), and the Games graph with parameters (729,112,1,20).

math.CO↗
arXiv2010

Bernstein type inequality in monotone rational approximation

The following analog of Bernstein inequality for monotone rational functions is established: if $R$ is an increasing on $[-1,1]$ rational function of degree $n$, then $$ R'(x)<\frac{9^n}{1-x^2}\|R\|,\quad x\in (-1,1). $$ The exponential dependence of constant factor on $n$ is shown, with sharp estimates for odd rational functions.

math.NA↗
arXiv2008

New asymptotic estimates for spherical designs

Let N(n, t) be the minimal number of points in a spherical t-design on the unit sphere S^n in R^{n+1}. For each n >= 3, we prove a new asymptotic upper bound N(n, t) <= C(n)t^{a_n}, where C(n) is a constant depending only on n, a_3 <= 4, a_4 <= 7, a_5 <= 9, a_6 <= 11, a_7 <= 12, a_8 <= 16, a_9 <= 19, a_10 <= 22, and a_n < n/2*log_2(2n), n > 10.

math.NA↗