An arc in a Steiner system S(2,4,v) of order v is a point set that contains at most two elements from each block. The largest size of an arc in S is denoted by \alpha_2(S) The upper bound \alpha_2(S)<=(v+2)/3 is always valid; an arc with (v+2)/3 points is called a maximum arc. We present a recursive construction with the following properties: (i) From two systems S'=S(2,4,v') and S''=S(2,4,v'') of respective orders v' and v'' an S= S(2,4,v) of order v=(1/3)(v'-1)(v''-1)+1. is obtained. (ii) \alpha_2(S)-1>=(\alpha_2(S')-1)(\alpha_2(S'')-1); in particular if both S' and S'' contain maximum arcs, then so does S, too. (iii) If each of S' and S'' can be covered with three maximum arcs incident with a common point then so does S too.

### A class of Steiner systems S(2,4,v) with arcs of extremal size

#### Abstract

An arc in a Steiner system S(2,4,v) of order v is a point set that contains at most two elements from each block. The largest size of an arc in S is denoted by \alpha_2(S) The upper bound \alpha_2(S)<=(v+2)/3 is always valid; an arc with (v+2)/3 points is called a maximum arc. We present a recursive construction with the following properties: (i) From two systems S'=S(2,4,v') and S''=S(2,4,v'') of respective orders v' and v'' an S= S(2,4,v) of order v=(1/3)(v'-1)(v''-1)+1. is obtained. (ii) \alpha_2(S)-1>=(\alpha_2(S')-1)(\alpha_2(S'')-1); in particular if both S' and S'' contain maximum arcs, then so does S, too. (iii) If each of S' and S'' can be covered with three maximum arcs incident with a common point then so does S too.
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Arc, Steiner system, Extremal arc
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11769/14737
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