2,319 research outputs found
The largest set partitioned by a subfamily of a cover
Define [lambda](n) to be the largest integer such that for each set A of size n and cover J of A, there exist B [subset of or equal to] A and G [subset of or equal to] J such that |B| = [lambda](n) and the restriction of G to B is a partition of B. It is shown that when n [ges] 3. The lower bound is proved by a probabilistic method. A related probabilistic algorithm for finding large sets partitioned by a subfamily of a cover is presented.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28503/1/0000300.pd
Colouring set families without monochromatic k-chains
A coloured version of classic extremal problems dates back to Erd\H{o}s and
Rothschild, who in 1974 asked which -vertex graph has the maximum number of
2-edge-colourings without monochromatic triangles. They conjectured that the
answer is simply given by the largest triangle-free graph. Since then, this new
class of coloured extremal problems has been extensively studied by various
researchers. In this paper we pursue the Erd\H{o}s--Rothschild versions of
Sperner's Theorem, the classic result in extremal set theory on the size of the
largest antichain in the Boolean lattice, and Erd\H{o}s' extension to
-chain-free families.
Given a family of subsets of , we define an
-colouring of to be an -colouring of the sets without
any monochromatic -chains . We
prove that for sufficiently large in terms of , the largest
-chain-free families also maximise the number of -colourings. We also
show that the middle level, , maximises the
number of -colourings, and give asymptotic results on the maximum
possible number of -colourings whenever is divisible by three.Comment: 30 pages, final versio
Quantitative Tverberg, Helly, & Carath\'eodory theorems
This paper presents sixteen quantitative versions of the classic Tverberg,
Helly, & Caratheodory theorems in combinatorial convexity. Our results include
measurable or enumerable information in the hypothesis and the conclusion.
Typical measurements include the volume, the diameter, or the number of points
in a lattice.Comment: 33 page
Matchings, coverings, and Castelnuovo-Mumford regularity
We show that the co-chordal cover number of a graph G gives an upper bound
for the Castelnuovo-Mumford regularity of the associated edge ideal. Several
known combinatorial upper bounds of regularity for edge ideals are then easy
consequences of covering results from graph theory, and we derive new upper
bounds by looking at additional covering results.Comment: 12 pages; v4 has minor changes for publicatio
Finding a non-minority ball with majority answers
Suppose we are given a set of balls each colored
either red or blue in some way unknown to us. To find out some information
about the colors, we can query any triple of balls
. As an answer to such a query we obtain (the
index of) a {\em majority ball}, that is, a ball whose color is the same as the
color of another ball from the triple. Our goal is to find a {\em non-minority
ball}, that is, a ball whose color occurs at least times among the
balls. We show that the minimum number of queries needed to solve this
problem is in the adaptive case and in the
non-adaptive case. We also consider some related problems
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