43 research outputs found
A note on tilted Sperner families with patterns
Let and be two nonnegative integers with and . We call
a \textit{(p,q)-tilted Sperner family
with patterns on [n]} if there are no distinct with:
Long (\cite{L}) proved that the cardinality of a (1,2)-tilted Sperner family
with patterns on is
We improve and generalize this result, and prove that the cardinality of
every ()-tilted Sperner family with patterns on [] is Comment: 8 page
Forbidding intersection patterns between layers of the cube
A family is said to be an antichain
if for all distinct . A classic result
of Sperner shows that such families satisfy , which is easily seen to be best possible. One can
view the antichain condition as a restriction on the intersection sizes between
sets in different layers of . More generally one can ask,
given a collection of intersection restrictions between the layers, how large
can families respecting these restrictions be? Answering a question of Kalai,
we show that for most collections of such restrictions, layered families are
asymptotically largest. This extends results of Leader and the author.Comment: 16 page
Forbidding a Set Difference of Size 1
How large can a family \cal A \subset \cal P [n] be if it does not contain
A,B with |A\setminus B| = 1? Our aim in this paper is to show that any such
family has size at most \frac{2+o(1)}{n} \binom {n}{\lfloor n/2\rfloor }. This
is tight up to a multiplicative constant of . We also obtain similar results
for families \cal A \subset \cal P[n] with |A\setminus B| \neq k, showing that
they satisfy |{\mathcal A}| \leq \frac{C_k}{n^k}\binom {n}{\lfloor n/2\rfloor
}, where C_k is a constant depending only on k.Comment: 8 pages. Extended to include bound for families \cal A \subset \cal P
[n] satisfying |A\setminus B| \neq k for all A,B \in \cal
Set families with a forbidden pattern
A balanced pattern of order 2d is an element P ∈ {+, −}2d
, where
both signs appear d times. Two sets A, B ⊂ [n] form a P-pattern,
which we denote by pat(A, B) = P, if A△B = {j1, . . . , j2d} with 1 ≤
j1 < · · · < j2d ≤ n and {i ∈ [2d] : Pi = +} = {i ∈ [2d] : ji ∈ A \ B}.
We say A ⊂ P [n] is P-free if pat(A, B) ̸= P for all A, B ∈ A. We
consider the following extremal question: how large can a family
A ⊂ P [n] be if A is P-free?
We prove a number of results on the sizes of such families.
In particular, we show that for some fixed c > 0, if P is a
d-balanced pattern with d < c log log n then | A |= o(2
n
). We
then give stronger bounds in the cases when (i) P consists of d+
signs, followed by d− signs and (ii) P consists of alternating signs.
In both cases, if d = o(
√
n)then | A |= o(2
n
). In the case of (i), this
is tight
On some problems in extremal, probabilistic and enumerative combinatorics
This is a study of a small selection of problems from various areas of Combinatorics and Graph Theory, a fast developing field that provides a diverse spectrum of powerful tools with numerous applications to computer science, optimization theory and economics. In this thesis, we focus on extremal, probabilistic and enumerative problems in this field.
A central theorem in combinatorics is Sperner's Theorem, which determines the maximum size of a family \F\subseteq \P(n) that does not contain a -chain . Erd\H{o}s later extended this result and determined the largest family not containing a -chain . Erd\H{o}s and Katona and later Kleitman asked how many such chains must appear in families whose size is larger than the corresponding extremal result. In Chapter 2 we answer their question for all families of size at most (1-\eps)2^n, provided is sufficiently larger compared to and \eps.
The result of Chapter 2 is an example of a supersaturation, or Erd\H{o}s--Rademacher type result, which seeks to answer how many forbidden objects must appear in a set whose size is larger than the corresponding result. These supersaturation results are a key ingredient to a very recently discovered proof method, called the Container method. Chapters 3 and 4 show various examples of this method in action. In Chapter 3 we, among others, give tight bounds on the logarithm of the number of -error correcting codes and illustrate how the Container method can be used to prove random analoges of classical extremal results. In Chapter 4 we solve a conjecture of Burosch--Demetrovics--Katona--Kleitman--Sapozhenko about estimating the number of families in which do not contain two sets and their union.
In Chapter 5 we improve an old result of Erd\H{o}s and Spencer. Folkman's theorem asserts that for each , there exists a natural number such that whenever the elements of are two-colored, there exists a set of size with the property that all the sums of the form , where is a nonempty subset of , are contained in and have the same color. In 1989, Erd\H{o}s and Spencer showed that , where is an absolute constant; here, we improve this bound significantly by showing that for all .
Fox--Grinshpun--Pach showed that every -coloring of the complete graph on vertices without a rainbow triangle contains a clique of size which uses at most two colors, and this bound is tight up to the constant factor. We show that if instead of looking for large cliques one only tries to find subgraphs of large chromatic number, one can do much better. In Chapter 6 we show, amongst others, that every such coloring contains a -colored subgraph with chromatic number at least , and this is best possible. As a direct corollary of our result we obtain a generalisation of the celebrated theorem of Erd\H{o}s-Szekeres, which states that any sequence of numbers contains a monotone subsequence of length at least
Digital Graphic Documentation and Architectural Heritage: Deformations in a 16th-Century Ceiling of the Pinelo Palace in Seville (Spain)
Suitable graphic documentation is essential to ascertain and conserve architectural heritage.
For the first time, accurate digital images are provided of a 16th-century wooden ceiling, composed
of geometric interlacing patterns, in the Pinelo Palace in Seville. Today, this ceiling suffers from
significant deformation. Although there are many publications on the digital documentation of
architectural heritage, no graphic studies on this type of deformed ceilings have been presented. This
study starts by providing data on the palace history concerning the design of geometric interlacing
patterns in carpentry according to the 1633 book by López de Arenas, and on the ceiling consolidation
in the 20th century. Images were then obtained using two complementary procedures: from a 3D
laser scanner, which offers metric data on deformations; and from photogrammetry, which facilitates
the visualisation of details. In this way, this type of heritage is documented in an innovative graphic
approach, which is essential for its conservation and/or restoration with scientific foundations and
also to disseminate a reliable digital image of the most beautiful ceiling of this Renaissance palace in
southern Europe.Instituto Universitario
de Arquitectura y Ciencias de la Construcción (IUACC) of the VII Plan Propio de Investigación y
Transferencia in the University of Sevill