Recent Developments in Combinatorial Geometry

Over a span of fifty years Paul Erdős has written many articles with this or a similar title. His countless results, which were obtained by the application of combinatorial and counting (random) methods, and the many deep problems raised and popularized in these papers, generated much research in combinatorics and graph theory. They played an important role in the emergence of a number of new areas in mathematics. One of these is combinatorial geometry, the study of extremal problems about finite arrangements of points, lines, circles, etc.

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[58]  Frank Harary,et al.  On the euclidean dimension of a wheel , 1988, Graphs Comb..

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[70]  P. Erdös,et al.  On Some Applications of Graph Theory to Geometry , 1967, Canadian Journal of Mathematics.

[71]  Fan Chung.  Sphere-and-point incidence relations in high dimensions with applications to unit distances and furthest-neighbor pairs , 1989 .

[72]  Hiroshi Maehara.  On the euclidean dimension of a complete multipartite graph , 1988, Discret. Math..

[73]  Herbert Edelsbrunner,et al.  A lower bound on the number of unit distances between the vertices of a convex polygon , 1991, J. Comb. Theory, Ser. A.

[74]  P. Erdös.  On Sets of Distances of n Points , 1946 .

[75]  J. Pach,et al.  An upper bound on the number of planar k-sets , 1989, 30th Annual Symposium on Foundations of Computer Science.

[76]  Paul Erdös,et al.  Repeated distances in space , 1988, Graphs Comb..

[77]  Katalin Vesztergombi,et al.  On the distribution of distances in finite sets in the plane , 1985, Discret. Math..

[78]  Yakov Shimeon Kupitz.  On Pairs of Disjoint Segments in Convex Position in The Plane , 1984 .

[79]  Frank Harary,et al.  On double and multiple interval graphs , 1979, J. Graph Theory.

[80]  N. Alon,et al.  Disjoint Simplices and Geometric Hypergraphs , 1989 .

[81]  J. Pach.  Decomposition of multiple packing and covering , 1980 .

[82]  József Beck,et al.  On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry , 1983, Comb..

[83]  Micha Sharir,et al.  Repeated Angles in the Plane and Related Problems , 1992, J. Comb. Theory, Ser. A.

[84]  Ernst Steinitz.  Über isoperimetrische Probleme bei konvexen Polyedern. , 1927 .

[85]  Noga Alon,et al.  Separating Pairs of Points by Standard Boxes , 1985, Eur. J. Comb..

[86]  W. Moser,et al.  On the Number of Ordinary Lines Determined by n Points , 1958, Canadian Journal of Mathematics.

[87]  Fan Chung,et al.  The number of different distances determined by n points in the plane , 1984 .

[88]  Endre Szemerédi,et al.  The number of different distances determined by a set of points in the Euclidean plane , 1992, Discret. Comput. Geom..

[89]  Peter Frankl,et al.  The Johnson-Lindenstrauss lemma and the sphericity of some graphs , 1987, J. Comb. Theory B.