Abstract
At most how many edges can an ordered graph ofn vertices have if it does not contain a fixed forbidden ordered subgraphH? It is not hard to give an asymptotically tight answer to this question, unlessH is a bipartite graph in which every vertex belonging to the first part precedes all vertices belonging to the second. In this case, the question can be reformulated as an extremal problem for zero-one matrices avoiding a certain pattern (submatrix)P. We disprove a general conjecture of Füredi and Hajnal related to the latter problem, and replace it by some weaker alternatives. We verify our conjectures in a few special cases whenP is the adjacency matrix of an acyclic graph and discuss the same question when the forbidden patterns are adjacency matrices of cycles. Our results lead to a new proof of the fact that the number of times that the unit distance can occur amongn points in the plane isO(n 4/3).
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Research supported by grants from NSF, BSF, PSC-CUNY, and OTKA (Hungar. Natonal Res. Found).
Research supported by OTKA (Hungar. Natonal Res. Found.) grants T-046234, AT-048826 and NK-62321.
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Pach, J., Tardos, G. Forbidden paths and cycles in ordered graphs and matrices. Isr. J. Math. 155, 359–380 (2006). https://doi.org/10.1007/BF02773960
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DOI: https://doi.org/10.1007/BF02773960