A theorem of arithmetic and its proof by Euler L.

By Euler L.

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1975). The number of unit distances in the plane. In: A. Hajnal et al. ), Infinite and Finite Sets, Vol. 2, pp. 939 -950. Colloquia Mathematica Societatis JBnos Bolyai vol. 10, North Holland. H. and Tardos, G. (2004). Note on distinct sums and distinct distances. In: J. ), Towards a Theory of Geometric Graphs, pp. 119- 126. 342, American Mathematical Society, Providence, RI. J. T. (1989). Finding squares and rectangles in sets of points. In: M. ), Graph-Theoretic Concepts in Computer Science, pp.

Question (3) on the minimum number of distinct patterns is largely open. For two-element patterns, we obtain higher-dimensional versions of the Problem of Distinct Distances. , by a cubic section of the d-dimensional integer lattice. The general lower bound of fl(nlld) was observed already in Erdijs (1946). 564)(Solymosi and Vu, 2005). This extends, in the same way as in the planar case, to a bound of fl(nk-2+2/d-2/(d(df 2))) for the minimum number of distinct k-point patterns of an n-element set, but even for triangles, nothing better is known.

And Thomas, R. (1997). The four-colour theorem. Journal of Combinatorial Theory, Series B, 7O(l):2-44. Soltks, L. (1991). Transmission in graphs: A bound and vertex removing. Mathematica Slovaca, 41(1):ll- 16. Chapter 2 PROBLEMS AND RESULTS ON GEOMETRIC PATTERNS Peter Brass JBnos Pach Abstract 1. Many interesting problems in combinatorial and computational geometry can be reformulated as questions about occurrences of certain patterns in finite point sets. We illustrate this framework by a few typical results and list a number of unsolved problems.

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