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Anti-mitre Steiner triple systems. (English) Zbl 0815.05017
A \((k,\ell)\)-configuration in a Steiner triple system \((V,B)\), is a subset of \(\ell\) triples from \(B\) whose union is a \(k\)-element subset of \(V\). The Pasch configuration is the \((6,4)\)-configuration on a set \(\{a,b, c,d, e,f\}\) with triples \(abe\), \(acf\), \(bdf\), \(cde\). The mitre is the \((7,5)\)-configuration on a set \(\{a,b, c,d, e,f, g\}\) with triples \(abe\), \(acf\), \(adg\), \(bcd\), \(efg\). A Steiner triple system (STS) is anti- Pasch (anti-mitre) if it does not contain any Pasch (mitre) configuration. Moreover, an STS is called \(r\)-sparse if every set of \(r+2\) elements carries fewer than \(r\) triples. Every STS is 3-sparse, is 4-sparse if and only if it is anti-Pasch, and 5-sparse if and only if it is both anti-Pasch and anti-mitre.
This paper makes substantial progress toward characterizing those \(v\) for which there exists an anti-mitre STS of order \(v\), and shows that for at least \({9 \over {16}}\) of the admissible values of \(v\) there exists an anti-mitre STS. The paper includes a table summarising small cyclic STS up to order 57, with the number of cyclic STS which are anti-Pasch or anti-mitre, or both (5-sparse). Also cyclic 5-sparse \(\text{STS} (v)\) are given for orders \(v=19\) and (\(v\equiv 1\) or \(3\pmod 6\)), \(33\leq v\leq 97\). This leads to the conjecture made in the paper that a 5-sparse \(\text{STS} (v)\) exists for all \(v\equiv 1, 3\pmod 6\), \(v\geq 33\). Existence of 5-sparse STS for orders 21, 25, 27 and 31 remains open, while not a single example of a 6-sparse \(\text{STS} (v)\) is currently known.

05B07 Triple systems
Full Text: DOI
[1] Brouwer, A.E.: Steiner triple systems without forbidden subconfigurations, Mathematisch Centrum Amsterdam, ZW 104–77, 1977 · Zbl 0367.05011
[2] Colbourn, M.J., Mathon, R.: On cyclic Steiner 2-designs, Ann. Discrete Math.7, 215–253 (1980) · Zbl 0438.05012
[3] Frenz, T.C., Kreher, D.L.: An algorithm for enumerating distinct cyclic Steiner systems, J. Comb. Math. Comb. Comput.11, 23–32 (1992) · Zbl 0755.05009
[4] Griggs, T.S., Murphy, J., Phelan, J.S.: Anti-Pasch Steiner triple systems, J. Comb. Inf. Syst. Sci.15, 79–84 (1990) · Zbl 0741.05009
[5] Lefmann, H., Phelps, K.T., Rödl, V.: Extremal problems for triple systems, J. Combinat. Designs1, 379–394 (1993) · Zbl 0817.05015
[6] Mathon, R., Phelps, K.T., Rosa, A.: Small Steiner triple systems and their properties, Ars Comb.15, 3–110 (1983) · Zbl 0516.05010
[7] Robinson, R.M.: The structure of certain triple systems, Math. Comput.20, 223–241 (1975) · Zbl 0293.05015
[8] Rosa, A.: Algebraic properties of designs and recursive constructions, Congressus Numer.13, 183–200 (1975)
[9] Stinson, D.R., Wei, R.: Some results on quadrilaterals in Steiner triple systems, Discrete Math.105, 207–219 (1992) · Zbl 0783.05022
[10] Street, A.P., Street, DJ.: The Combinatorics of Experimental Design, Clarendon Press, Oxford, 1987 · Zbl 0622.05001
[11] Teirlinck, L.: Large sets of disjoint designs and related structures, in: Contemporary Design Theory (J.H. Dinitz and D.R. Stinson, eds.) Wiley, New York, 1992, pp. 561–592 · Zbl 0805.05012
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