arXiv subjects
Publications and source records attributed to Bartłomiej Bychawski.
We show that there exists $c > 0$ such that any subset of $\{1,\ldots,N\}$ having size $\gg N / \exp( (\log\log\log N)^c )$ contains a nontrivial pattern of the form $x,x+y,x+2y,x+y^3$. It is the first configuration of complexity strictly greater than $0$, other than refinements of arithmetic progressions, for which quantitative bounds over integers were obtained.
We study sums of higher even powers in the coordinate rings of singular planar curves $x^M=y^m$ for coprime positive integers $m 1$, can be of codimension $2$, contrary to the quadratic case.
In this paper we prove that for circulants of squarefree orders Wilson's conjecture hold, that is each nontrivially unstable circulant of such order has Wilson type. We show that actually only criteria (C.1) and (C.4) are needed.
It is known that the canonical double cover of any connected nonbipartite graph have an automorphism group of the form $H \rtimes \mathbb{Z}_2$, where $H$ is the set of automorphism which preserve bipartite parts. We construct connected nonbipartite and vertex determining graphs whose canonical double covers have auromorphisms group isomorphic to any semisimple product of $\mathbb{Z}_2$ with any abstract group $H$. Later we show, that the canonical double cover of any asymmetric graph have abelian automorphisms group of odd order. The above construction provides an example of asymmetric graph for any such group. By modifying the aforementioned construction we obtain graphs which have any possible number and type of graphs with isomorphic double covers.