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- W2022037033 abstract "Let t >1, let P 1 ,…, P t be pairwise disjoint nonvoid subsets of a finite set P , and let L be a collection of subsets of P called lines. We say that (P 1 ,…,P t ; L ) is a colored incidence structure if (i) each line meets at least two blocks P i and P j , (ii) for arbitrary distinct xϵP i and yϵP j the pair { x , y } is a subset of exactly one line if i ≠ j and at most one line if i = j . Extending the work of de Bruijn and Erdös and Meshulam we show that for a coloured incidence structure (P 1 ,…,P t ; L ) with | P 1 |⩽…⩽| P t | in general | L | exceeds 1+| P 1 |+…+ | P t −1 |. The exceptional cases are exactly: (i) | L |=|P 1 |+·+|P t−1 | if the structur e is a truncated projective plane and (ii) | L |=1+|P 1 |+·+∣P t−1 | holds in exactly six cases: a projective plane, the dual of an affine plane, the dual of a modification of an affine plane, a near-pencil, a structure with 2 lines and a 9-element structure with 7 lines." @default.
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- W2022037033 date "1988-01-01" @default.
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- W2022037033 title "Multicolored lines in a finite geometry" @default.
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- W2022037033 doi "https://doi.org/10.1016/0012-365x(88)90068-4" @default.
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