86 research outputs found
Chromatic Numbers of Simplicial Manifolds
Higher chromatic numbers of simplicial complexes naturally
generalize the chromatic number of a graph. In any fixed dimension
, the -chromatic number of -complexes can become arbitrarily
large for [6,18]. In contrast, , and only
little is known on for .
A particular class of -complexes are triangulations of -manifolds. As a
consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number
of any fixed surface is finite. However, by combining results from the
literature, we will see that for surfaces becomes arbitrarily large
with growing genus. The proof for this is via Steiner triple systems and is
non-constructive. In particular, up to now, no explicit triangulations of
surfaces with high were known.
We show that orientable surfaces of genus at least 20 and non-orientable
surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a
projective Steiner triple systems, we construct an explicit triangulation of a
non-orientable surface of genus 2542 and with face vector
that has 2-chromatic number 5 or 6. We also give orientable examples with
2-chromatic numbers 5 and 6.
For 3-dimensional manifolds, an iterated moment curve construction [18] along
with embedding results [6] can be used to produce triangulations with
arbitrarily large 2-chromatic number, but of tremendous size. Via a topological
version of the geometric construction of [18], we obtain a rather small
triangulation of the 3-dimensional sphere with face vector
and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio
Short proofs of some extremal results
We prove several results from different areas of extremal combinatorics,
giving complete or partial solutions to a number of open problems. These
results, coming from areas such as extremal graph theory, Ramsey theory and
additive combinatorics, have been collected together because in each case the
relevant proofs are quite short.Comment: 19 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
Analytic methods for uniform hypergraphs
This paper develops analityc methods for investigating uniform hypergraphs.
Its starting point is the spectral theory of 2-graphs, in particular, the
largest and the smallest eigenvalues of 2-graphs. On the one hand, this simple
setup is extended to weighted r-graphs, and on the other, the
eigenvalues-numbers are generalized to eigenvalues-functions, which encompass
also other graph parameters like Lagrangians and number of edges. The resulting
theory is new even for 2-graphs, where well-settled topics become challenges
again. The paper covers a multitude of topics, with more than a hundred
concrete statements to underpin an analytic theory for hypergraphs. Essential
among these topics are a Perron-Frobenius type theory and methods for extremal
hypergraph problems. Many open problems are raised and directions for possible
further research are outlined.Comment: 71 pages. Corrected wrong claim in the introductio
Large cliques or cocliques in hypergraphs with forbidden order-size pairs
The well-known ErdĆs-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 ErdĆs-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
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