Abstract
Hill [6] showed that the largest cap in PG(5,3) has cardinality 56. Using this cap it is easy to construct a cap of cardinality 45 in AG(5,3). Here we show that the size of a cap in AG(5,3) is bounded above by 48. We also give an example of three disjoint 45-caps in AG(5,3). Using these two results we are able to prove that the Steiner triple system AG(5,3) is 6-chromatic, and so we exhibit the first specific example of a 6-chromatic Steiner triple system.
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Bruen, A., Haddad, L. & Wehlau, D. Caps and Colouring Steiner Triple Systems. Designs, Codes and Cryptography 13, 51–55 (1998). https://doi.org/10.1023/A:1008293805734
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DOI: https://doi.org/10.1023/A:1008293805734