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11 - Dimension and Cut Vertices: An Application of Ramsey Theory
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- By William T. Trotter, School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA, Bartosz Walczak, Theoretical Computer Science Department, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków 30-348, Poland, Ruidong Wang, Blizzard Entertainment, Irvine, CA 92618, USA
- Edited by Steve Butler, Iowa State University, Joshua Cooper, University of South Carolina, Glenn Hurlbert, Virginia Commonwealth University
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- Book:
- Connections in Discrete Mathematics
- Published online:
- 25 May 2018
- Print publication:
- 14 June 2018, pp 187-199
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Summary
Abstract
Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every, if P is a poset and the dimension of a subposet B of P is at most d whenever the cover graph of B is a block of the cover graph of P, then the dimension of P is at most d + 2.We also construct examples that show that this inequality is best possible. We consider the proof of the upper bound to be fairly elegant and relatively compact. However, we know of no simple proof for the lower bound, and our argument requires a powerful tool known as the Product Ramsey Theorem. As a consequence, our constructions involve posets of enormous size.
Introduction
We assume that the reader is familiar with basic notation and terminology for partially ordered sets (here we use the short term posets), including chains and antichains, minimal and maximal elements, linear extensions, order diagrams, and cover graphs. Extensive background information on the combinatorics of posets can be found in [17, 18].
We will also assume that the reader is familiar with basic concepts of graph theory, including the following terms: connected and disconnected graphs, components, cut vertices, and k-connected graphs for an integer. Recall that when G is a connected graph, a connected induced subgraph H of G is called a block of G when H is 2-connected and there is no subgraph of G which contains H as a proper subgraph and is also 2-connected.
Here are the analogous concepts for posets. A poset P is said to be connected if its cover graph is connected. A subposet B of P is said to be convex if y ∈ B whenever x, z ∈ B and x < y < z in P. Note that when B is a convex subposet of P, the cover graph of B is an induced subgraph of the cover graph of P. A convex subposet B of P is called a component of P when the cover graph of B is a component of the cover graph of P. A convex subposet B of P is called a block of P when the cover graph of B is a block in the cover graph of P.
Ramsey Theory and Sequences of Random Variables
- WILLIAM T. TROTTER, PETER WINKLER
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- Journal:
- Combinatorics, Probability and Computing / Volume 7 / Issue 2 / June 1998
- Published online by Cambridge University Press:
- 01 June 1998, pp. 221-238
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We consider probability spaces which contain a family {EA[ratio ]A⊆{1, 2, …, n}, [mid ]A[mid ]=k} of events indexed by the k-element subsets of {1, 2, …, n}. A pair (A, B) of k-element subsets of {1, 2, …, n} is called a shift pair if the largest k−1 elements of A coincide with the smallest k−1 elements of B. For a shift pair (A, B), Pr[AB¯] is the probability that event EA is true and EB is false. We investigate how large the minimum value of Pr[AB¯], taken over all shift pairs, can be. As n→∞, this value converges to a number λk, with ½−1/2k+2[les ]λk[les ] ½−1/4k+2. We show that λk is a strictly increasing function of k, with λ1=¼ and λ2=1/3.
For k=1, our results have the following natural interpretation. If a fair coin is tossed repeatedly, and event Ei is true when the ith toss is heads, then for all i and j with i<j, Pr[EiĒj]=¼. Furthermore, as we show in this paper, for any ε>0, there is an n such that for any sequence E1, E2, …, En of events in an arbitrary probability space, there are indices i<j with Pr[EiĒj]<¼+ε. The results and techniques we develop in this research, together with further applications of Ramsey theory, are then used to show that the supremum of fractional dimensions of interval orders is exactly 4, answering a question of Brightwell and Scheinerman.
Generalizing the ¼+ε result to random variables X1, X2, …, Xn with values in an m-element set, we obtain a finite version of de Finetti's theorem without the exchangeability hypothesis: for any fixed m, k and ε, every sufficiently long sequence of such random variables has a length-k subsequence at variation distance less than ε from an i.i.d. mix.
New Perspectives on Interval Orders and Interval Graphs
- Edited by R. A. Bailey, Queen Mary University of London
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- Book:
- Surveys in Combinatorics, 1997
- Published online:
- 29 March 2010
- Print publication:
- 31 July 1997, pp 237-286
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Summary
Summary Interval orders and interval graphs are particularly natural examples of two widely studied classes of discrete structures: partially ordered sets and undirected graphs. So it is not surprising that researchers in such diverse fields as mathematics, computer science, engineering and the social sciences have investigated structural, algorithmic, enumerative, combinatorial, extremal and even experimental problems associated with them. In this article, we survey recent work on interval orders and interval graphs, including research on on-line coloring, dimension estimates, fractional parameters, balancing pairs, hamiltonian paths, ramsey theory, extremal problems and tolerance orders. We provide an outline of the arguments for many of these results, especially those which seem to have a wide range of potential applications. Also, we provide short proofs of some of the more classical results on interval orders and interval graphs. Our goal is to provide fresh insights into the current status of research in this area while suggesting new perspectives and directions for the future.
Introduction
A complex process (manufacturing computer chips, for example) is often broken into a series of tasks, each with a specified starting and ending time. Task A precedes Task B if A ends before B begins. When A precedes B, the output of A can safely be used as input to B, and resources dedicated to the completion of A, such as machines or personnel, can now be applied to B. When A and B have overlapping time periods, they may be viewed as conflicting tasks, in the sense that they compete for limited resources.