The maximum number of halving lines and the rectilinear crossing number of Kn for n

Abstract For n ⩽ 27 we present exact values for the maximum number h ( n ) of halving lines and h ˜ ( n ) of halving pseudolines, determined by n points in the plane. For this range of values of n we also present exact values of the rectilinear c r ¯ ( K n ) and the pseudolinear c r ˜ ( K n ) crossing numbers of the complete graph K n . h ˜ ( n ) and c r ˜ ( K n ) are new for n ∈ { 14 , 16 , 18 , 20 , 22 , 23 , 24 , 25 , 26 , 27 } , h ( n ) is new for n ∈ { 16 , 18 , 20 , 22 , 23 , 24 , 25 , 26 , 27 } , and c r ¯ ( K n ) is new for n ∈ { 20 , 22 , 23 , 24 , 25 , 26 , 27 } .

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