Multicolored lines in a finite geometry (Q1112829)

Let P be a set of points, partitioned into blocks \(P_ 1,...,P_ t\), and let \underbar{L} be a set of lines (subsets of P) satisfying the following conditions: (i) each line intersects at least two of the blocks (ii) any two points in distinct blocks are on a unique line (iii) any two points in the same block are on at most one line. Then \((P_ 1,...,P_ t;\underline{L})\) is called colored incidence structure. (One may think of this as the structure induced by a (partial) coloration of a finite linear space, where one discards colorless and monochromatic lines.) Extending a result of \textit{R. Meshulam} [``On multicolored lines in a linear space'', J. Comb. Theory, Ser. A 40, 150-155 (1985; Zbl 0616.05003)] the authors show that one always has \(| \underline{L}| \geq | P_ 1| +...+| P_{t-1}|\) and that in general even \(| \underline{L}| >| P_ 1| +...+| P_{t-1}|.\) They completely classify the cases where \(| L| =(1+)| P_ 1| +...+| P_{t-1}|;\) one obtains (truncated) projective planes, (modified) dual affine planes, near-pencils - a structure with 2 lines and a structure on 9 points with 7 lines.