14 research outputs found
On the Erd\H{o}s-Tuza-Valtr Conjecture
The Erd\H{o}s-Szekeres conjecture states that any set of more than
points in the plane with no three on a line contains the vertices of a convex
-gon. Erd\H{o}s, Tuza, and Valtr strengthened the conjecture by stating that
any set of more than points in a
plane either contains the vertices of a convex -gon, points lying on a
concave downward curve, or points lying on a concave upward curve. They
also showed that the generalization is actually equivalent to the
Erd\H{o}s-Szekeres conjecture.
We prove the first new case of the Erd\H{o}s-Tuza-Valtr conjecture since the
original 1935 paper of Erd\H{o}s and Szekeres. Namely, we show that any set of
points in the plane with no three points on a line and no
two points sharing the same -coordinate either contains 4 points lying on a
concave downward curve or the vertices of a convex -gon.Comment: 16 pages, 8 figure
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 combinatorial classification of nondegenerate configurations in the plane
AbstractWe classify nondegenerate plane configurations by attaching, to each such configuration of n points, a periodic sequence of permutations of {1, 2, …, n} which satisfies some simple conditions; this classification turns out to be appropriate for questions involving convexity. In 1881 Perrin stated that every sequence satisfying these conditions was the image of some plane configuration. We show that this statement is incorrect by exhibiting a counterexample, for n = 5, and prove that for n ⩽ 5 every sequence essentially distinct from this one is realized geometrically by giving a complete classification of configurations in these cases; there is 1 combinatorial equivalence class for n = 3, 2 for n = 4, and 19 for n = 5. We develop some basic notions of the geometry of “allowable sequences” in the course of proving this classification theorem. Finally, we state some results and an open problem on the realizability question in the general case
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete