21 research outputs found
Recent trends in Euclidean Ramsey theory
AbstractWe give a brief summary of several new results in Euclidean Ramsey theory, a subject which typically investigates properties of configurations in Euclidean space which are preserved under finite partitions of the space
Approximate Euclidean Ramsey theorems
According to a classical result of Szemer\'{e}di, every dense subset of
contains an arbitrary long arithmetic progression, if is large
enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson
says that every dense subset of contains an arbitrary large
grid, if is large enough. Here we generalize these results for separated
point sets on the line and respectively in the Euclidean space: (i) every dense
separated set of points in some interval on the line contains an
arbitrary long approximate arithmetic progression, if is large enough. (ii)
every dense separated set of points in the -dimensional cube in
\RR^d contains an arbitrary large approximate grid, if is large enough. A
further generalization for any finite pattern in \RR^d is also established.
The separation condition is shown to be necessary for such results to hold. In
the end we show that every sufficiently large point set in \RR^d contains an
arbitrarily large subset of almost collinear points. No separation condition is
needed in this case.Comment: 11 pages, 1 figure
Simplices and Regular Polygonal Tori in Euclidean Ramsey Theory
We show that any finite affinely independent set can be isometrically
embedded into a regular polygonal torus, that is, a finite product of regular
polygons. As a consequence, with a straightforward application of
K\v{r}\'{i}\v{z}'s theorem, we get an alternative proof of the fact that all
finite affinely independent sets are Ramsey, a result which was originally
proved by Frankl and R\"{o}dl.Comment: 7 pages; corrected typo
The density of sets containing large similar copies of finite sets
Funding: VK is supported by the Croatian Science Foundation, project n◦ UIP-2017-05-4129 (MUNHANAP). AY is supported by the Swiss National Science Foundation, grant n◦ P2SKP2 184047.We prove that if E⊆Rd (d≥2) is a Lebesgue-measurable set with density larger than n−2n−1, then E contains similar copies of every n-point set P at all sufficiently large scales. Moreover, 'sufficiently large' can be taken to be uniform over all P with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of n-point sets tends to 1 at a rate 1−O(n−1/5log n).PreprintPeer reviewe