19 research outputs found
On constants in the Füredi–Hajnal and the Stanley–Wilf conjecture
AbstractFor a given permutation matrix P, let fP(n) be the maximum number of 1-entries in an n×n (0,1)-matrix avoiding P and let SP(n) be the set of all n×n permutation matrices avoiding P. The Füredi–Hajnal conjecture asserts that cP:=limn→∞fP(n)/n is finite, while the Stanley–Wilf conjecture asserts that sP:=limn→∞|SP(n)|n is finite.In 2004, Marcus and Tardos proved the Füredi–Hajnal conjecture, which together with the reduction introduced by Klazar in 2000 proves the Stanley–Wilf conjecture.We focus on the values of the Stanley–Wilf limit (sP) and the Füredi–Hajnal limit (cP). We improve the reduction and obtain sP⩽2.88cP2 which decreases the general upper bound on sP from sP⩽constconstO(klog(k)) to sP⩽constO(klog(k)) for any k×k permutation matrix P. In the opposite direction, we show cP=O(sP4.5).For a lower bound, we present for each k a k×k permutation matrix satisfying cP=Ω(k2)
Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns
We prove that the Stanley-Wilf limit of any layered permutation pattern of
length is at most , and that the Stanley-Wilf limit of the
pattern 1324 is at most 16. These bounds follow from a more general result
showing that a permutation avoiding a pattern of a special form is a merge of
two permutations, each of which avoids a smaller pattern. If the conjecture is
true that the maximum Stanley-Wilf limit for patterns of length is
attained by a layered pattern then this implies an upper bound of for
the Stanley-Wilf limit of any pattern of length .
We also conjecture that, for any , the set of 1324-avoiding
permutations with inversions contains at least as many permutations of
length as those of length . We show that if this is true then the
Stanley-Wilf limit for 1324 is at most
On the structure of graphs with forbidden induced substructures
One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints.
In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs.
Erdős and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every -edge-colouring of the complete graph on vertices there is a monochromatic clique on at least vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs.
In the second part of this thesis we focus more on order-size pairs; an order-size pair is the family consisting of all graphs of order and size , i.e. on vertices with edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs , i.e. for approaching infinity, the limit superior of the fraction of all possible sizes , such that the order-size pair does not avoid the pair
Compact Representation for Matrices of Bounded Twin-Width
For every fixed , we design a data structure that
represents a binary matrix that is -twin-ordered. The data
structure occupies bits, which is the least one could hope for, and
can be queried for entries of the matrix in time per query.Comment: 24 pages, 2 figure
On ordered Ramsey numbers of matchings versus triangles
For graphs and with linearly ordered vertex sets, the \ordered
Ramsey number is the smallest positive integer such that any
red-blue coloring of the edges of the complete ordered graph on
vertices contains either a blue copy of or a red copy of . Motivated
by a problem of Conlon, Fox, Lee, and Sudakov (2017), we study the numbers
where is an ordered matching on vertices.
We prove that almost all -vertex ordered matchings with interval
chromatic number 2 satisfy and
, improving a recent result by Rohatgi (2019).
We also show that there are -vertex ordered matchings with interval
chromatic number at least 3 satisfying , which asymptotically matches the best known lower bound on these
off-diagonal ordered Ramsey numbers for general -vertex ordered matchings.Comment: 16 pages, 2 figures; extended abstract to appear at EuroComb 202
The Landscape of Bounds for Binary Search Trees
Binary search trees (BSTs) with rotations can adapt to various kinds of structure in search sequences, achieving amortized access times substantially better than the Theta(log n) worst-case guarantee. Classical examples of structural properties include static optimality, sequential access, working set, key-independent optimality, and dynamic finger, all of which are now known to be achieved by the two famous online BST algorithms (Splay and Greedy). (...) In this paper, we introduce novel properties that explain the efficiency of sequences not captured by any of the previously known properties, and which provide new barriers to the dynamic optimality conjecture. We also establish connections between various properties, old and new. For instance, we show the following. (i) A tight bound of O(n log d) on the cost of Greedy for d-decomposable sequences. The result builds on the recent lazy finger result of Iacono and Langerman (SODA 2016). On the other hand, we show that lazy finger alone cannot explain the efficiency of pattern avoiding sequences even in some of the simplest cases. (ii) A hierarchy of bounds using multiple lazy fingers, addressing a recent question of Iacono and Langerman. (iii) The optimality of the Move-to-root heuristic in the key-independent setting introduced by Iacono (Algorithmica 2005). (iv) A new tool that allows combining any finite number of sound structural properties. As an application, we show an upper bound on the cost of a class of sequences that all known properties fail to capture. (v) The equivalence between two families of BST properties. The observation on which this connection is based was known before - we make it explicit, and apply it to classical BST properties. (...