10 research outputs found
A note on lower bounds for hypergraph Ramsey numbers
We improve upon the lower bound for 3-colour hypergraph Ramsey numbers,
showing, in the 3-uniform case, that The old bound, due to Erd\H{o}s and Hajnal, was Comment: 6 page
Hypergraph Ramsey numbers of cliques versus stars
Let denote the complete -uniform hypergraph on vertices
and the -uniform hypergraph on vertices consisting of all
edges incident to a given vertex. Whereas many hypergraph Ramsey
numbers grow either at most polynomially or at least exponentially, we show
that the off-diagonal Ramsey number exhibits an
unusual intermediate growth rate, namely, for some positive
constants and . The proof of these bounds brings in a novel Ramsey
problem on grid graphs which may be of independent interest: what is the
minimum such that any -edge-coloring of the Cartesian product contains either a red rectangle or a blue ?Comment: 13 page
Large cliques or co-cliques in hypergraphs with forbidden order-size pairs
The well-known Erd\H{o}s-Hajnal conjecture states that for any graph ,
there exists such that every -vertex graph that contains no
induced copy of has a homogeneous set of size at least . We
consider a variant of the Erd\H{o}s-Hajnal problem for hypergraphs where we
forbid a family of hypergraphs described by their orders and sizes. For graphs,
we observe that if we forbid induced subgraphs on vertices and edges
for any positive and , then we obtain large
homogeneous sets. For triple systems, in the first nontrivial case , for
every , we give bounds on the minimum size of a
homogeneous set in a triple system where the number of edges spanned by every
four vertices is not in . In most cases the bounds are essentially tight. We
also determine, for all , whether the growth rate is polynomial or
polylogarithmic. Some open problems remain.Comment: A preliminary version of this manuscript appeared as arXiv:2303.0957
Large cliques or cocliques in hypergraphs with forbidden order-size pairs
The well-known Erdős-Hajnal conjecture states that for any graph , there exists such that every -vertex graph that contains no induced copy of has a homogeneous set of size at least . We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on vertices and edges for any positive and , then we obtain large homogeneous sets. For triple systems, in the first nontrivial case , for every , we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in . In most cases the bounds are essentially tight. We also determine, for all , whether the growth rate is polynomial or polylogarithmic. Some open problems remain
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
THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References
and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a