### 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 xε{lunate}P_{i} and yε{lunate}P_{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}|+·+{divides}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.

Original language | English |
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Pages (from-to) | 149-163 |

Number of pages | 15 |

Journal | Discrete Mathematics |

Volume | 71 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1988 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*71*(2), 149-163. https://doi.org/10.1016/0012-365X(88)90068-4