9 research outputs found
On equivalence between maximal and maximum antichains over range-restricted vectors with integral coordinates and the number of such maximal antichains
Consider a strict partially ordered set consisting of all
-dimensional vectors with integral coordinates restricted in a certain
range. We found that any maximal antichain is also maximum, and the maximum
size has a simple expression in terms of the range. Properties of the number of
maximal antichains given the range are explored. We present our proof, the
application on other areas, and some open questions.Comment: Submitted partial result to Journal Orde
Almost all permutation matrices have bounded saturation functions
Saturation problems for forbidden graphs have been a popular area of research
for many decades, and recently Brualdi and Cao initiated the study of a
saturation problem for 0-1 matrices. We say that 0-1 matrix is saturating
for the forbidden 0-1 matrix if avoids but changing any zero to a
one in creates a copy of . Define to be the minimum possible
number of ones in an 0-1 matrix that is saturating for . Fulek
and Keszegh proved that for every 0-1 matrix , either or
. They found two 0-1 matrices for which , as well as infinite families of 0-1 matrices for which . Their results imply that for almost all 0-1 matrices .
Fulek and Keszegh conjectured that there are many more 0-1 matrices such
that besides the ones they found, and they asked for a
characterization of all permutation matrices such that .
We affirm their conjecture by proving that almost all permutation
matrices have . We also make progress on the
characterization problem, since our proof of the main result exhibits a family
of permutation matrices with bounded saturation functions
Degrees of nonlinearity in forbidden 0–1 matrix problems
AbstractA 0–1 matrix A is said to avoid a forbidden 0–1 matrix (or pattern) P if no submatrix of A matches P, where a 0 in P matches either 0 or 1 in A. The theory of forbidden matrices subsumes many extremal problems in combinatorics and graph theory such as bounding the length of Davenport–Schinzel sequences and their generalizations, Stanley and Wilf’s permutation avoidance problem, and Turán-type subgraph avoidance problems. In addition, forbidden matrix theory has proved to be a powerful tool in discrete geometry and the analysis of both geometric and non-geometric algorithms.Clearly a 0–1 matrix can be interpreted as the incidence matrix of a bipartite graph in which vertices on each side of the partition are ordered. Füredi and Hajnal conjectured that if P corresponds to an acyclic graph then the maximum weight (number of 1s) in an n×n matrix avoiding P is O(nlogn). In the first part of the article we refute of this conjecture. We exhibit n×n matrices with weight Θ(nlognloglogn) that avoid a relatively small acyclic matrix. The matrices are constructed via two complementary composition operations for 0–1 matrices. In the second part of the article we simplify one aspect of Keszegh and Geneson’s proof that there are infinitely many minimal nonlinear forbidden 0–1 matrices. In the last part of the article we investigate the relationship between 0–1 matrices and generalized Davenport–Schinzel sequences. We prove that all forbidden subsequences formed by concatenating two permutations have a linear extremal function
Sharp Bounds on Davenport-Schinzel Sequences of Every Order
One of the longest-standing open problems in computational geometry is to
bound the lower envelope of univariate functions, each pair of which
crosses at most times, for some fixed . This problem is known to be
equivalent to bounding the length of an order- Davenport-Schinzel sequence,
namely a sequence over an -letter alphabet that avoids alternating
subsequences of the form with length
. These sequences were introduced by Davenport and Schinzel in 1965 to
model a certain problem in differential equations and have since been applied
to bounding the running times of geometric algorithms, data structures, and the
combinatorial complexity of geometric arrangements.
Let be the maximum length of an order- DS sequence over
letters. What is asymptotically? This question has been answered
satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and
Nivasch) when is even or . However, since the work of Agarwal,
Sharir, and Shor in the mid-1980s there has been a persistent gap in our
understanding of the odd orders.
In this work we effectively close the problem by establishing sharp bounds on
Davenport-Schinzel sequences of every order . Our results reveal that,
contrary to one's intuition, behaves essentially like
when is odd. This refutes conjectures due to Alon et al.
(2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the
Symposium on Computational Geometry, 201
Linear bound on extremal functions of some forbidden patterns in 0-1 matrices
AbstractIn this note by saying that a 0–1 matrix A avoids a pattern P given as a 0–1 matrix we mean that no submatrix of A either equals P or can be transformed into P by replacing some 1 entries with 0 entries. We present a new method for estimating the maximal number of the 1 entries in a matrix that avoids a certain pattern. Applying this method we give a linear bound on the maximal number ex(n,L1) of the 1 entries in an n by n matrix avoiding pattern L1 and thereby we answer the question that was asked by Gábor Tardos. Furthermore, we use our approach on patterns related to L1