International Symposium on Graph Drawing and Network Visualization

Graph Drawing and Network Visualization pp 207-216

# On the Zarankiewicz Problem for Intersection Hypergraphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

## Abstract

Let d and t be fixed positive integers, and let $$K^d_{t,\ldots ,t}$$ denote the complete d-partite hypergraph with t vertices in each of its parts, whose hyperedges are the d-tuples of the vertex set with precisely one element from each part. According to a fundamental theorem of extremal hypergraph theory, due to Erdős [7], the number of hyperedges of a d-uniform hypergraph on n vertices that does not contain $$K^d_{t,\ldots ,t}$$ as a subhypergraph, is $$n^{d-\frac{1}{t^{d-1}}}$$. This bound is not far from being optimal.

We address the same problem restricted to intersection hypergraphs of $$(d-1)$$-dimensional simplices in $$\mathbb {R}^d$$. Given an n-element set $$\mathcal {S}$$ of such simplices, let $$\mathcal {H}^d(\mathcal {S})$$ denote the d-uniform hypergraph whose vertices are the elements of $$\mathcal {S}$$, and a d-tuple is a hyperedge if and only if the corresponding simplices have a point in common. We prove that if $$\mathcal {H}^d(\mathcal {S})$$ does not contain $$K^d_{t,\ldots ,t}$$ as a subhypergraph, then its number of edges is O(n) if $$d=2$$, and $$O(n^{d-1+\epsilon })$$ for any $$\epsilon >0$$ if $$d \ge 3$$. This is almost a factor of n better than Erdős’s above bound. Our result is tight, apart from the error term $$\epsilon$$ in the exponent.

In particular, for $$d=2$$, we obtain a theorem of Fox and Pach [8], which states that every $$K_{t,t}$$-free intersection graph of nsegments in the plane has O(n) edges. The original proof was based on a separator theorem that does not generalize to higher dimensions. The new proof works in any dimension and is simpler: it uses size-sensitive cuttings, a variant of random sampling. We demonstrate the flexibility of this technique by extending the proof of the planar version of the theorem to intersection graphs of x-monotone curves.

### References

1. 1.
Agarwal, P.K., Matousek, J., Sharir, M.: On range searching with semialgebraic sets. II. SIAM J. Comput. 42(6), 2039–2062 (2013)
2. 2.
Bollobás, B.: Modern Graph Theory. Springer (1998)Google Scholar
3. 3.
Brown, W.G.: On graphs that do not contain a Thomsen graph. Canad. Math. Bull. 9, 281285 (1966)
4. 4.
Chazelle, B., Edelsbrunner, H., Guibas, L.J., Sharir, M., Snoeyink, J.: Computing a face in an arrangement of line segments and related problems. SIAM J. Comput. 22(6), 1286–1302 (1993)
5. 5.
Clarkson, K., Edelsbrunner, H., Guibas, L., Sharir, M., Welzl, E.: Combinatorial complexity bounds for arrangements of curves and spheres. Discrete Comput. Geom. 5, 99–160 (1990)
6. 6.
de Berg, M., Schwarzkopf, O.: Cuttings and applications. Int. J. Comput. Geom. Appl. 5(4), 343–355 (1995)
7. 7.
Erdös, P.: On extremal problems of graphs and generalized graphs. Israel J. Math 2, 183–190 (1964)
8. 8.
Fox, J., Pach, J.: Separator theorems and Turán-type results for planar intersection graphs. Adv. Math. 219(3), 1070–1080 (2008)
9. 9.
Fox, J., Pach, J.: A separator theorem for string graphs and its applications. Comb. Probab. Comput. 19(3), 371–390 (2010)
10. 10.
Fox, J., Pach, J.: Applications of a new separator theorem for string graphs. Comb., Probab. Comput. 23, 66–74 (2014)
11. 11.
Fox, J., Pach, J., Sheffer, A., Suk, A., Zahl, J.: A semi-algebraic version of Zarankiewicz’s problem. ArXiv e-prints (2014)Google Scholar
12. 12.
Haussler, D., Welzl, E.: Epsilon-nets and simplex range queries. Discrete Comput. Geom. 2, 127–151 (1987)
13. 13.
Matoušek, J.: Lectures in Discrete Geometry. Springer, New York (2002)
14. 14.
Kővári, T., Sós, V.T., Turán, P.: On a problem of K. Zarankiewicz. Colloquium Math. 3, 50–57 (1954)Google Scholar
15. 15.
Pach, J., Sharir, M.: On planar intersection graphs with forbidden subgraphs. J. Graph Theory 59(3), 205–214 (2008)
16. 16.
Pellegrini, M.: On counting pairs of intersecting segments and off-line triangle range searching. Algorithmica 17(4), 380–398 (1997)
17. 17.
Radoic̆ić, R., Tóth, G.: The discharging method in combinatorial geometry and the Pach-Sharir conjecture. In: Goodman, J.E., Pach, J., Pollack, J. (eds.) Proceedings of the Joint Summer Research Conference on Discrete and Computational Geometry, vol. 453, pp. 319–342. Contemporary Mathematics, AMS (2008)Google Scholar
18. 18.
Reiman, I.: Uber ein problem von K. Zarankiewicz. Acta Mathematica Academiae Scientiarum Hungarica 9, 269–273 (1958)
19. 19.
Tagansky, B.: A new technique for analyzing substructures in arrangements of piecewise linear surfaces. Discrete Comput. Geom. 16(4), 455–479 (1996)