Equal entries in totally positive matrices

We show that the maximal number of equal entries in a totally positive (resp. totally nonsingular) n-by-nn-by-n matrix is Θ(n4/3)Θ(n4/3) (resp. Θ(n3/2)Θ(n3/2)). Relationships with point-line incidences in the plane, Bruhat order of permutations, and TP   completability are also presented. We also examine the number and positionings of equal 2-by-22-by-2 minors in a 2-by-n2-by-nTP   matrix, and give a relationship between the location of equal 2-by-22-by-2 minors and outerplanar graphs.