54 research outputs found
Semi-algebraic and semi-linear Ramsey numbers
An -uniform hypergraph is semi-algebraic of complexity
if the vertices of correspond to points in
, and the edges of are determined by the sign-pattern of
degree- polynomials. Semi-algebraic hypergraphs of bounded complexity
provide a general framework for studying geometrically defined hypergraphs.
The much-studied semi-algebraic Ramsey number
denotes the smallest such that every -uniform semi-algebraic hypergraph
of complexity on vertices contains either a clique of size
, or an independent set of size . Conlon, Fox, Pach, Sudakov, and Suk
proved that R_{r}^{\mathbf{t}}(n,n)<\mbox{tw}_{r-1}(n^{O(1)}), where
\mbox{tw}_{k}(x) is a tower of 2's of height with an on the top. This
bound is also the best possible if is sufficiently large with
respect to . They conjectured that in the asymmetric case, we have
for fixed . We refute this conjecture by
showing that for some
complexity .
In addition, motivated by results of Bukh-Matou\v{s}ek and
Basit-Chernikov-Starchenko-Tao-Tran, we study the complexity of the Ramsey
problem when the defining polynomials are linear, that is, when . In
particular, we prove that , while
from below, we establish .Comment: 23 pages, 1 figur
Hasse diagrams with large chromatic number
For every positive integer , we construct a Hasse diagram with
vertices and chromatic number , which significantly improves
on the previously known best constructions of Hasse diagrams having chromatic
number . In addition, if we also require that our Hasse diagram
has girth at least , we can achieve a chromatic number of at least
.
These results have the following surprising geometric consequence. They imply
the existence of a family of curves in the plane such that
the disjointness graph of is triangle-free (or have high
girth), but the chromatic number of is polynomial in . Again, the
previously known best construction, due to Pach, Tardos and T\'oth, had only
logarithmic chromatic number.Comment: 11 pages, 1 figur
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