A well-known problem in Design Theory is the study of the possible existence of blocking sets in Steiner systems. In this paper, we introduce the concept of perfect blocking sets in G-designs and determine all the possible v for which there exist P 3-designs having perfect blocking sets.
Let Σ = (X, B) a 4-cycle system of order v = 1 + 8k. A c-colouring of type s is a map φ∶ B → C, with C set of colours, such that exactly c colours are used and for every vertex x all the blocks containing x are coloured exactly with s colours. Let 4k = qs + r, with q, r ≥ 0. φ is equitable if for every vertex x the set of the 4k blocks containing x is parted in r colour classes of cardinality q + 1 and s − r colour classes of cardinality q. In this paper we study tricolourings, for which s = 3, with the hypothesis that either v ≡ 9 mod 24 or v ≡ 17 mod 24, determining an upper bound for c.
In this paper, we completely determine the spectrum of edge balanced H-designs, where H is a 3-uniform hypergraph with 2 or 3 edges, such that H has strong chromatic number χs(H)=3.
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