On multi-dimensional patterns

Kitaev, Sergey and Robbins, Jakayla (2007) On multi-dimensional patterns. Pure Mathematics and Applications, 18 (3-4). pp. 291-299. ISSN 1218-4586

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Abstract

We generalize the concept of pattern occurrence in permutations, words or matrices to that in n-dimensional objects, which are basically sets of (n + 1)-tuples. In the case n = 3, we give a possible interpretation of such patterns in terms of bipartite graphs. For zero-box patterns we study vanishing borders related to bipartite Ramsey problems in the case of two dimensions. Also, we study the maximal number of 1’s in binary objects avoiding (in two different senses) a zero-box pattern.