5,964 research outputs found
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
Lines in Euclidean Ramsey theory
Let be a sequence of points on a line with consecutive points of
distance one. For every natural number , we prove the existence of a
red/blue-coloring of containing no red copy of and no
blue copy of for any . This is best possible up to the
constant in the exponent. It also answers a question of Erd\H{o}s, Graham,
Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every
natural number , there is a set and a
red/blue-coloring of containing no red copy of and no
blue copy of .Comment: 7 page
Chromatic number of Euclidean plane
If the chromatic number of Euclidean plane is larger than four, but it is
known that the chromatic number of planar graphs is equal to four, then how
does one explain it? In my opinion, they are contradictory to each other. This
idea leads to confirm the chromatic number of the plane about its exact value
On metric Ramsey-type phenomena
The main question studied in this article may be viewed as a nonlinear
analogue of Dvoretzky's theorem in Banach space theory or as part of Ramsey
theory in combinatorics. Given a finite metric space on n points, we seek its
subspace of largest cardinality which can be embedded with a given distortion
in Hilbert space. We provide nearly tight upper and lower bounds on the
cardinality of this subspace in terms of n and the desired distortion. Our main
theorem states that for any epsilon>0, every n point metric space contains a
subset of size at least n^{1-\epsilon} which is embeddable in Hilbert space
with O(\frac{\log(1/\epsilon)}{\epsilon}) distortion. The bound on the
distortion is tight up to the log(1/\epsilon) factor. We further include a
comprehensive study of various other aspects of this problem.Comment: 67 pages, published versio
On a problem of Erd\H{o}s and Rothschild on edges in triangles
Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by
H(N,C), such that every N-vertex graph with at least CN^2 edges, each of which
is contained in at least one triangle, must contain an edge that is in at least
H(N,C) triangles. In particular, Erd\H{o}s asked in 1987 to determine whether
for every C>0 there is \epsilon >0 such that H(N,C) > N^\epsilon, for all
sufficiently large N. We prove that H(N,C) = N^{O(1/log log N)} for every fixed
C < 1/4. This gives a negative answer to the question of Erd\H{o}s, and is best
possible in terms of the range for C, as it is known that every N-vertex graph
with more than (N^2)/4 edges contains an edge that is in at least N/6
triangles.Comment: 8 page
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