Counting Families of Mutually Intersecting Sets

@article{Brouwer2013CountingFO,
  title={Counting Families of Mutually Intersecting Sets},
  author={Andries E. Brouwer and C. F. Mills and W. H. Mills and A. Verbeek},
  journal={Electron. J. Comb.},
  year={2013},
  volume={20},
  pages={8}
}
We determine the number of maximal intersecting families on a 9-set and nd 423295099074735261880. We determine the number of independent sets of the Kneser graph K(9; 4) and nd 366996244568643864340. We determine the number of intersecting families on an 8-set and on a 9-set and nd 14704022144627161780744368338695925293142507520 and 12553242487940503914363982718112298267975272720808010757809032705650591023015520462677475328 (roughly 1:255 10 91 ), respectively. 
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References

SHOWING 1-10 OF 19 REFERENCES
Voting Fairly: Transitive Maximal Intersecting Families of Sets
TLDR
The enumeration of the 207,650,662,008 maximal families of intersecting subsets of X whose group of symmetries is transitive for |X|<13.
The order dimension of the complete graph
A computation of the eighth Dedekind number
We compute the eighth Dedekind number, or the number of monotone collections of subsets of a set with eight elements. The number obtained is 56, 130, 437, 228, 687, 557, 907, 788.
On counting independent sets in sparse graphs
TLDR
Two results are proved concerning approximate counting of independent sets in graphs with constant maximum degree $\Delta$ that imply that the Markov chain Monte Carlo technique is likely to fail and that no fully polynomial randomized approximation scheme can exist for $\Delta \geq 25$ unless $\mathrm{RP}=\mathrm(NP)$.
On Dedekind’s problem: The number of monotone Boolean functions
with an = ce~nli, /3„ = e'(Iog w)/«1'2.The number \p(n) is equal to the number of ideals, or of antichains,or of monotone increasing functions into 0 and 1 definable on thelattice of subsets of an
Problems and Results in Combinatorial Analysis
I gave many lectures by this and similar titles, many in fact in these conferences and I hope in my lecture in 1978 I will give a survey of the old problems and describe what happened to them. In the
Stable Winning Coalitions
  • D. Loeb
  • Computer Science, Economics
  • 1996
We introduce the notion of a stable winning coalition in a multiplayer game as a new system of classification of games. An axiomatic refinement of this classification for three-player games is also
The Art of Computer Programming
TLDR
The arrangement of this invention provides a strong vibration free hold-down mechanism while avoiding a large pressure drop to the flow of coolant fluid.
Counting families of mutually intersecting sets, internal report
  • ZN 41, Afd. Zuivere Wiskunde, Math. Centrum,
  • 1972
On the number of clique Boolean functions
...
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