35 research outputs found
A counterexample to a geometric Hales-Jewett type conjecture
P\'or and Wood conjectured that for all there exists
with the following property: whenever points, no of which are
collinear, are chosen in the plane and each of them is assigned one of
colours, then there must be a line (that is, a maximal set of collinear points)
all of whose points have the same colour. The conjecture is easily seen to be
true for (by the pigeonhole principle) and in the case it is an
immediate corollary of the Motzkin-Rabin theorem. In this note we show that the
conjecture is false for .Comment: 4 pages, 2 figures; updated title, added figures and more detail
Combinatorial and Additive Number Theory Problem Sessions: '09--'19
These notes are a summary of the problem session discussions at various CANT
(Combinatorial and Additive Number Theory Conferences). Currently they include
all years from 2009 through 2019 (inclusive); the goal is to supplement this
file each year. These additions will include the problem session notes from
that year, and occasionally discussions on progress on previous problems. If
you are interested in pursuing any of these problems and want additional
information as to progress, please email the author. See
http://www.theoryofnumbers.com/ for the conference homepage.Comment: Version 3.4, 58 pages, 2 figures added 2019 problems on 5/31/2019,
fixed a few issues from some presenters 6/29/201
Forbidden vector-valued intersections
We solve a generalised form of a conjecture of Kalai motivated by attempts to
improve the bounds for Borsuk's problem. The conjecture can be roughly
understood as asking for an analogue of the Frankl-R\"odl forbidden
intersection theorem in which set intersections are vector-valued. We discover
that the vector world is richer in surprising ways: in particular, Kalai's
conjecture is false, but we prove a corrected statement that is essentially
best possible, and applies to a considerably more general setting. Our methods
include the use of maximum entropy measures, VC-dimension, Dependent Random
Choice and a new correlation inequality for product measures.Comment: 40 page
Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory
The goal of this present manuscript is to introduce the reader to the
nonstandard method and to provide an overview of its most prominent
applications in Ramsey theory and combinatorial number theory.Comment: 126 pages. Comments welcom
Some interesting problems
A ≤W B. (This refers to Wadge reducible.) Answer: The first question was answered by Hjorth [83] who showed that it is independent. 1.2 A subset A ⊂ ω ω is compactly-Γ iff for every compact K ⊂ ω ω we have that A ∩ K is in Γ. Is it consistent relative to ZFC that compactly-Σ 1 1 implies Σ 1 1? (see Miller-Kunen [111], Becker [11]) 1.3 (Miller [111]) Does ∆ 1 1 = compactly- ∆ 1 1 imply Σ 1 1 = compactly-Σ 1 1? 1.4 (Prikry see [62]) Can L ∩ ω ω be a nontrivial Σ 1 1 set? Can there be a nontrivial perfect set of constructible reals? Answer: No, for first question Velickovic-Woodin [192]. question Groszek-Slaman [71]. See also Gitik [67]
Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory
The aim of this Arbeitsgemeinschaft was to introduce young researchers with various backgrounds to the multifaceted and mutually perpetuating connections between ergodic theory, topological dynamics, combinatorics, and number theory
Recommended from our members
Topics in metric geometry, combinatorial geometry, extremal combinatorics and additive combinatorics
In this thesis, we consider several combinatorial topics, belonging to the areas appearing in the thesis title. Given a non-empty complete metric space , a family of continuous maps is a \emph{contractive family} if there exists and , we construct a subset for some , such that , while . (Here and .) We also prove some extensions of this result. Among other ingredients, the proof also includes an application of a quantitative equidistribution result for polynomials. In the final part, we consider the Graham-Pollak problem for hypergraphs. Let be the minimum number of complete -partite -graphs needed to partition the edge set of the complete -uniform hypergraph on vertices. We disprove a conjecture that , by showing that . The proof is based on the relationship between this problem and a problem about decomposing products of complete graphs, and understanding how the Graham-Pollak theorem (for graphs) affects what can happen here.I would like to thank Trinity College and Department of Pure Mathematics
and Mathematical Statistics for their generous financial support and hospitality during PhD studies