# The Dimension of Random Graph Orders

• Béla Bollobás
• Graham Brightwell
Part of the Algorithms and Combinatorics book series (AC, volume 14)

## Summary

The random graph order P n, p is obtained from a random graph G n, p on [n] by treating an edge between vertices i and j, with i ≺ j in [n], as a relation i < j, and taking the transitive closure. This paper forms part of a project to investigate the structure of the random graph order P n, p throughout the range of p = p(n). We give bounds on the dimension of P n, p for various ranges. We prove that, if p log log n → ∞ and ε > 0 then, almost surely,
$$\left( {1 - \in } \right)\sqrt {\frac{{\log n}} {{\log (1/q)}}} \leqslant \dim P_{n,p} \leqslant (1 + \in )\sqrt {\frac{{4\log n}} {{3\log (1/q)}}} .$$
We also prove that there are constants c 1, c 2 such that, if p log n → 0 and p ≥ log n/n, then
$$c_1 p^{ - 1} \leqslant \dim P_{n,p} \leqslant c_2 p^{ - 1} .$$

We give some bounds for various other ranges of p(n), but several questions are left open.

### Keywords

Expense milO aCnE

## Preview

### References

1. 1.
M. Albert and A. Frieze, Random graph orders, Order 6 (1989) 19–30.
2. 2.
N. Alon, B. Bollobás, G. Brightwell and S. Janson, Linear extensions of a random partial order, Annals of Applied Prob. 4 (1994) 108–123.
3. 3.
A. Barak and P. Erdős, On the maximal number of strongly independent vertices in a random acyclic directed graph, SIAM J. Algebraic and Disc. Methods 5 (1984) 508–514.
4. 4.
B. Bollobás, Random Graphs, Academic Press, London, 1985, xv+447pp.
5. 5.
B.Bollobás and G.Brightwell, The width of random graph orders, submitted.Google Scholar
6. 6.
B.Bollobás and G.Brightwell, The structure of random graph orders, submitted.Google Scholar
7. 7.
B. Bollobás and I. Leader, Isoperimetric inequalities and fractional set systems, J. Combinatorial Theory (A) 56 (1991) 63–74.
8. 8.
G. Brightwell, Models of random partial orders, in Surveys in Combinatorics 1993, Invited papers at the 14th British Combinatorial Conference, K. Walker Ed., Cambridge University Press (1993).Google Scholar
9. 9.
P. Erdős, H. Kierstead and W. T. Trotter, The dimension of random ordered sets, Random Structures and Algorithms 2 (1991) 253–275.
10. 10.
P. Erdős and A. Renyi, On random graphs I, Publ. Math. Debrecen 6 (1959) 290–297.
11. 11.
P. Erdős and A. Renyi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960) 17–61.Google Scholar
12. 12.
Z. Fiiredi and J. Kahn, On the dimensions of ordered sets of bounded degree, Order 3 (1986) 17–20.Google Scholar
13. 13.
M. Hall, Combinatorial Theory, 2nd Edn., Wiley-Interscience Series in Discrete Mathematics (1986) xv+440pp.Google Scholar
14. 14.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th Edn., Oxford University Press (1979) xvi+426pp.
15. 15.
I. Leader, Discrete isoperimetric inequalities, in Probabilistic Combinatorics and its Applications, Proceedings of Symposia in Applied Mathematics 44, American Mathematical Society, Providence (1991).Google Scholar
16. 16.
C. M. Newman, Chain lengths in certain directed graphs, Random Structures and Algorithms 3 (1992) 243–253.
17. 17.
C. M. Newman and J. E. Cohen, A stochastic theory of community food webs: IV. Theory of food chain lengths in large webs, Proc. R. Soc. London Ser. B 228 (1986) 355–377.
18. 18.
K. Simon, D. Crippa and F. Collenberg, On the distribution of the transitive closure in random acyclic digraphs, Lecture Notes in Computer Science 726 (1993) 345–356.Google Scholar
19. 19.
W. T. Trotter, Inequalities in dimension theory for posets, Proc. Amer. Math. Soc. 47 (1975) 311–316.
20. 20.
W. T. Trotter, Embedding finite posets in cubes, Discrete Math. 12 (1975) 165–172.
21. 21.
W. T. Trotter, Combinatorics and Partially Ordered Sets: Dimension Theory, The Johns Hopkins University Press, Baltimore (1992) xiv+307pp