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Extremal problems in discrete geometry
TLDR
Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.
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Combinatorics and Partially Ordered Sets: Dimension Theory
Primarily intended for research mathematicians and computer scientists, Combinatorics and Partially Ordered Sets: Dimension Theory also serves as a useful text for advanced students in either field.…
Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures
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The Ramsey number of a graph with bounded maximum degree
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Unit distances in the Euclidean plane
Tolerance graphs
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An on-line graph coloring algorithm with sublinear performance ratio
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The Order Dimension of Convex Polytopes
TLDR
The order dimension of the partially ordered set P is exactly 4 for every convex polytope M and the subposet of P determined by the vertices and faces is critical in the sense that deleting any element leaves a poset of dimension 3.
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Explicit matchings in the middle levels of the Boolean lattice
New classes of explicit matchings for the bipartite graph ℬ(k) consisting of the middle two levels of the Boolean lattice on 2k+1 elements are constructed and counted. This research is part of an…
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The Order Dimension of Planar Maps
TLDR
If {\bf M} is any planar map, then the order dimension of P, the poset formed by taking the vertices, edges, and faces of a 3-connected planarMap, is still at most 4.
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