##
**\(k\)-common consequents in Boolean matrices.**
*(English)*
Zbl 0879.04001

Let \(Q\) be a binary relation on an \(n\)-element set \(\{a_1,\dots ,a_n\}\) and \(M(Q)\) be its incidence matrix \((m_{ij})\), i.e. \(m_{ij}=1\) if \((a_i,a_j)\in Q\) and \(m_{ij}=0\) otherwise. A subset \(\{a_{i_1},\dots , a_{i_r}\}\) has a common consequent if the rows of \(M(Q)\) corresponding to \(a_{i_1},\dots ,a_{i_r}\) have a 1 in the same \(k\)-th column. The least such \(k\) is denoted by \(L_Q(a_{i_1},\dots ,a_{i_r})\). We denote by \(L_Q(r)\) the maximum of \(L_Q(a_{i_1},\dots ,a_{i_r})\), where the elements run through all groups of cardinality \(r\) for which a common consequent exists. The values of \(L_Q(r)\) are estimated in the paper. The case \(L_Q(2)\) was solved previously by Š. Schwarz.

Reviewer: I.Chajda (Olomouc)

### MSC:

03E20 | Other classical set theory (including functions, relations, and set algebra) |

05C30 | Enumeration in graph theory |

### References:

[1] | Š. Schwarz: Common consequents in directed graphs. Czechoslovak Math. J. 35 (1985), no. 110, 212-246. · Zbl 0575.05041 |

[2] | R.A. Brualdi and Bolian Liu: Generalized exponents of primitive directed graphs. J. Graph Theory 14 (1990), no. 4, 483-499. · Zbl 0714.05028 · doi:10.1002/jgt.3190140413 |

[3] | Š. Schwarz: A combinatorial problem arising in finite Markov chains. Math. Slovaca 36 (1986), 21-28. · Zbl 0615.15006 |

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