A two-coloring inequality for Euclidean two-arrangements

Simmons, Gustavus J.; Wetzel, John E.

doi:10.1090/S0002-9939-1979-0539644-4

A two-coloring inequality for Euclidean two-arrangements
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by Gustavus J. Simmons and John E. Wetzel PDF

Proc. Amer. Math. Soc. 77 (1979), 124-127 Request permission

Abstract:

We prove that for any properly two-colored arrangement of lines in the Euclidean plane having, say, r red and g green regions with $r \geqslant g$, the inequality \[ r \leqslant 2g - 2 - \sum \limits _P {(\lambda (P) - 2)} \] holds, where for each point P of intersection of the lines, $\lambda (P)$ is the number of lines of the arrangement that contain P. This strengthens recent results of Simmons and Grünbaum.

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A linear partitioning of the plane which extends the known solutions of four combinatorial problems

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