A Unified Generalization of Two Results in Ordered Hypergraph Avoidance

The Füredi–Hajnal conjecture, originally posed in [6] and solved by Marcus and Tardos in [10], can be cast as an extremal problem on ordered graphs. It states that any n-vertex bipartite graph that avoids a fixed permutation graph has O(n) edges. Balogh, Bollobás and Morris [1] recently generalized this result to show that any n-vertex, ordered hypergraph that avoids a fixed permutation graph has O(n) edges. Independently, Klazar and Marcus [9] generalized the FürediHajnal conjecture in another direction. Extending the idea of a permutation graph to a d-permutation hypergraph, they showed that any n-vertex, d-uniform, d-partite, ordered hypergraph that avoids a fixed d-permutation hypergraph has O(nd−1) edges. This paper extends these two results into a single generalization. We adapt the techniques of [1], which are based upon the original FürediHajnal result, in order to use the generalized d-dimensional result of [9] as the base. Our main result implies a single generalization of all of the previously mentioned results: that any n-vertex, ordered hypergraph that avoids a fixed d-permutation hypergraph has O(nd−1) edges.