393 research outputs found
Boolean algebras and Lubell functions
Let denote the power set of . A collection
\B\subset 2^{[n]} forms a -dimensional {\em Boolean algebra} if there
exist pairwise disjoint sets , all non-empty
with perhaps the exception of , so that \B={X_0\cup \bigcup_{i\in I}
X_i\colon I\subseteq [d]}. Let be the maximum cardinality of a family
\F\subset 2^X that does not contain a -dimensional Boolean algebra.
Gunderson, R\"odl, and Sidorenko proved that where .
In this paper, we use the Lubell function as a new measurement for large
families instead of cardinality. The Lubell value of a family of sets \F with
\F\subseteq \tsupn is defined by h_n(\F):=\sum_{F\in \F}1/{{n\choose |F|}}.
We prove the following Tur\'an type theorem. If \F\subseteq 2^{[n]} contains
no -dimensional Boolean algebra, then h_n(\F)\leq 2(n+1)^{1-2^{1-d}} for
sufficiently large . This results implies , where is an absolute constant independent of and . As a
consequence, we improve several Ramsey-type bounds on Boolean algebras. We also
prove a canonical Ramsey theorem for Boolean algebras.Comment: 10 page
Threshold functions and Poisson convergence for systems of equations in random sets
We present a unified framework to study threshold functions for the existence
of solutions to linear systems of equations in random sets which includes
arithmetic progressions, sum-free sets, -sets and Hilbert cubes. In
particular, we show that there exists a threshold function for the property
" contains a non-trivial solution of
", where is a random set and each of
its elements is chosen independently with the same probability from the
interval of integers . Our study contains a formal definition of
trivial solutions for any combinatorial structure, extending a previous
definition by Ruzsa when dealing with a single equation.
Furthermore, we study the behaviour of the distribution of the number of
non-trivial solutions at the threshold scale. We show that it converges to a
Poisson distribution whose parameter depends on the volumes of certain convex
polytopes arising from the linear system under study as well as the symmetry
inherent in the structures, which we formally define and characterize.Comment: New version with minor corrections and changes in notation. 24 Page
Small union with large set of centers
Let be a fixed set. By a scaled copy of around
we mean a set of the form for some .
In this survey paper we study results about the following type of problems:
How small can a set be if it contains a scaled copy of around every point
of a set of given size? We will consider the cases when is circle or sphere
centered at the origin, Cantor set in , the boundary of a square
centered at the origin, or more generally the -skeleton () of an
-dimensional cube centered at the origin or the -skeleton of a more
general polytope of .
We also study the case when we allow not only scaled copies but also scaled
and rotated copies and also the case when we allow only rotated copies
Polyhedral Cones of Magic Cubes and Squares
Using computational algebraic geometry techniques and Hilbert bases of
polyhedral cones we derive explicit formulas and generating functions for the
number of magic squares and magic cubes.Comment: 14 page
Zonotopes and four-dimensional superconformal field theories
The a-maximization technique proposed by Intriligator and Wecht allows us to
determine the exact R-charges and scaling dimensions of the chiral operators of
four-dimensional superconformal field theories. The problem of existence and
uniqueness of the solution, however, has not been addressed in general setting.
In this paper, it is shown that the a-function has always a unique critical
point which is also a global maximum for a large class of quiver gauge theories
specified by toric diagrams. Our proof is based on the observation that the
a-function is given by the volume of a three dimensional polytope called
"zonotope", and the uniqueness essentially follows from Brunn-Minkowski
inequality for the volume of convex bodies. We also show a universal upper
bound for the exact R-charges, and the monotonicity of a-function in the sense
that a-function decreases whenever the toric diagram shrinks. The relationship
between a-maximization and volume-minimization is also discussed.Comment: 29 pages, 15 figures, reference added, typos corrected, version
published in JHE
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