Erdos once posed the following: Show that n non-collinear points in the real plane determine at least n connecting lines. This is a problem in projective geometry. The equivalent problem in affine geometry is the following: Show that n non-collinear points in the real affine plane determine at least n-1 distinct slopes. This problem was first considered by P. R. Scott and finally solved by Peter Ungar. After Ungar proved this conjecture, research has centered on two areas: finding and characterizing configurations achieving the minimum number of slopes, which are called slope critical configurations, and answering the question in other settings such as finite planes.
Current research in sum covers includes the following questions.
Those interested in this topic can contact either Mark Fitch or Robert Jamison.
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