Doilies with Vertex Sets 275, 286,... 462

In this paper we close the gap between the results of [8] and [9] by showing that there are 11-doilies for vertex sets of size 275; 286; : : : ; 462: In [8] it is shown that there are 11-doilies with vertex size 462; 473; : : : ; 1001; and in [9] it is shown that there are such diagrams with vertex size 231; 242; : : : ; 352: This is the third paper in a series showing that 11-doilies exist for any possible size of vertex set, answering a question of Gr unbaum. We continue to extend the method that is developed in [7], [8], and [9]. The crucial step in the method of those paper is based on saturated chain decompositions of planar, spanning subgraphs of the p-hypercube, and on edge-disjoint path decompositions of planar, spanning subgraphs of the p-hypercube. We continue to study these type of decompositions. In constructing the doilies we use a basic structure, called the Venn graph of a doodle of the doily. Using the new one that we create here we show the existence of the above mentioned diagrams. In fact, without checking all the details we show that there are at least 2 (non-isomorphic) 11-doilies with these vertex sets.