13,388 research outputs found
The Ramsey Theory of Henson graphs
Analogues of Ramsey's Theorem for infinite structures such as the rationals
or the Rado graph have been known for some time. In this context, one looks for
optimal bounds, called degrees, for the number of colors in an isomorphic
substructure rather than one color, as that is often impossible. Such theorems
for Henson graphs however remained elusive, due to lack of techniques for
handling forbidden cliques. Building on the author's recent result for the
triangle-free Henson graph, we prove that for each , the
-clique-free Henson graph has finite big Ramsey degrees, the appropriate
analogue of Ramsey's Theorem.
We develop a method for coding copies of Henson graphs into a new class of
trees, called strong coding trees, and prove Ramsey theorems for these trees
which are applied to deduce finite big Ramsey degrees. The approach here
provides a general methodology opening further study of big Ramsey degrees for
ultrahomogeneous structures. The results have bearing on topological dynamics
via work of Kechris, Pestov, and Todorcevic and of Zucker.Comment: 75 pages. Substantial revisions in the presentation. Submitte
Classical and consecutive pattern avoidance in rooted forests
Following Anders and Archer, we say that an unordered rooted labeled forest
avoids the pattern if in each tree, each sequence of
labels along the shortest path from the root to a vertex does not contain a
subsequence with the same relative order as . For each permutation
, we construct a bijection between -vertex
forests avoiding and
-vertex forests avoiding ,
giving a common generalization of results of West on permutations and
Anders--Archer on forests. We further define a new object, the forest-Young
diagram, which we use to extend the notion of shape-Wilf equivalence to
forests. In particular, this allows us to generalize the above result to a
bijection between forests avoiding and forests avoiding for . Furthermore, we give recurrences
enumerating the forests avoiding , , and other sets
of patterns. Finally, we extend the Goulden--Jackson cluster method to study
consecutive pattern avoidance in rooted trees as defined by Anders and Archer.
Using the generalized cluster method, we prove that if two length- patterns
are strong-c-forest-Wilf equivalent, then up to complementation, the two
patterns must start with the same number. We also prove the surprising result
that the patterns and are strong-c-forest-Wilf equivalent, even
though they are not c-Wilf equivalent with respect to permutations.Comment: 39 pages, 11 figure
Trees and Markov convexity
We show that an infinite weighted tree admits a bi-Lipschitz embedding into
Hilbert space if and only if it does not contain arbitrarily large complete
binary trees with uniformly bounded distortion. We also introduce a new metric
invariant called Markov convexity, and show how it can be used to compute the
Euclidean distortion of any metric tree up to universal factors
Reconciling taxonomy and phylogenetic inference: formalism and algorithms for describing discord and inferring taxonomic roots
Although taxonomy is often used informally to evaluate the results of
phylogenetic inference and find the root of phylogenetic trees, algorithmic
methods to do so are lacking. In this paper we formalize these procedures and
develop algorithms to solve the relevant problems. In particular, we introduce
a new algorithm that solves a "subcoloring" problem for expressing the
difference between the taxonomy and phylogeny at a given rank. This algorithm
improves upon the current best algorithm in terms of asymptotic complexity for
the parameter regime of interest; we also describe a branch-and-bound algorithm
that saves orders of magnitude in computation on real data sets. We also
develop a formalism and an algorithm for rooting phylogenetic trees according
to a taxonomy. All of these algorithms are implemented in freely-available
software.Comment: Version submitted to Algorithms for Molecular Biology. A number of
fixes from previous versio
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
Coherence for indexed symmetric monoidal categories
Indexed symmetric monoidal categories are an important refinement of
bicategories -- this structure underlies several familiar bicategories,
including the homotopy bicategory of parametrized spectra, and its equivariant
and fiberwise generalizations.
In this paper, we extend existing coherence theorems to the setting of
indexed symmetric monoidal categories. The most central theorem states that a
large family of operations on a bicategory defined from an indexed symmetric
monoidal category are all canonically isomorphic. As a part of this theorem, we
introduce a rigorous graphical calculus that specifies when two such operations
admit a canonical isomorphism.Comment: 100 pages, 64 figures, 13 table
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