Combinatorial computing approach to selected extremal problems in geometry

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Some of my Favourite Problems in Number Theory, Combinatorics, and Geometry

To the memor!l of m!l old friend Professor George Sved.I heard of his untimel!l death while writing this paper

On the Order Dimension of Convex Geometries

We study the order dimension of the lattice of closed sets for a convex geometry. Further, we prove the existence of large convex geometries realized by planar point sets that have very low order dimension. We show that the planar point set of Erdos and Szekeres from 1961 which is a set of 2^(n-2) points and contains no convex n-gon has order dimension n - 1 and any larger set of points has order dimension strictly larger than n - 1.Comment: 12 pages, 2 figure

Unavoidable patterns in complete simple topological graphs

In this paper, we show that every complete $n$-vertex simple topological graph contains a topological subgraph on at least $(\log n)^{1/4 - o(1)}$ vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This improves the previously known bound of $\Omega(\log^{1/8}n)$ due to Pach, Solymosi, and T\'oth. We also show that every complete $n$-vertex simple topological graph contains a planar path of length at least $(\log n)^{1 - o(1)}$

Enumeration of small triangle free Ramsey Graphs

In 1930, a paper by Frank Plumpton Ramsey entitled On a Problem of Formal Logic appeared in the Proceedings of the London Mathematical Society. Although the impetus of this paper was one of mathematical logic, a far reaching combinatorial result was needed by Ramsey to achieve his objective. This combinatorial result became known as Ram \sey\u27s Theorem. One of the combinatorial structures which was developed during the study of Ramsey\u27s Theorem is that of a Ramsey graph. A Ramsey graph, denoted (k,l,n,e), is defined as an undirected graph that contains no cliques of size k, no independent sets of size I, with order n, and size e. Knowledge of Ramsey graphs is useful in the improvement of bounds and sometimes the calculation of exact values for various Ramsey number parameter situations. Straightforward enumeration of (k, I, n, e) Ramsey graphs for larger values of n is intractable with the current computing technology available. In order to produce such graphs, specialized algorithms need to be implemented. This thesis provides the theoretical background developed by Graver and Yackel [GRA68a], expanded upon by Grinstead and Roberts [GRl82a], and generalized by Radziszowski and Kreher [RAD88a, RAD88b] for the implementation of algorithms utilized for the enumeration of various Ram sey graphs. An object oriented graph manipulation package, including the above mentioned Ramsey graph enumeration algorithms, is implemented and documented. This package is utilized for the enumeration of all (3,3), (3,4), (3,5) and (3, 6) graphs. Some (3, 7) and (3, 8) also are calculated. These results duplicate and verify Ramsey graphs previously enumerated during other investigations. [RAD88a, RAD88b] In addition to these results, some newly enumerated (3,8) critical graphs, as well as some newly enumerated (3,9) graphs, including a minimum (3, 9, 26, 52) -graph are presented

On List-Coloring and the Sum List Chromatic Number of Graphs.

This thesis explores several of the major results in list-coloring in an expository fashion. As a specialization of list coloring, the sum list chromatic number is explored in detail. Ultimately, the thesis is designed to motivate the discussion of coloring problems and, hopefully, interest the reader in the branch of coloring problems in graph theory

The Parameterized Complexity of Finding Point Sets with Hereditary Properties

We consider problems where the input is a set of points in the plane and an integer k, and the task is to find a subset S of the input points of size k such that S satisfies some property. We focus on properties that depend only on the order type of the points and are monotone under point removals. We exhibit a property defined by three forbidden patterns for which finding a k-point subset with the property is W[1]-complete and (assuming the exponential time hypothesis) cannot be solved in time n^{o(k/log k)}. However, we show that problems of this type are fixed-parameter tractable for all properties that include all collinear point sets, properties that exclude at least one convex polygon, and properties defined by a single forbidden pattern