The fine triangle intersection problem for \((K_4 - e)\)-designs. (English) Zbl 1292.05054
Summary: Let \(\mathrm{Fin}(v)=\{(s,t):\exists\) a pair of \((K4-e)\)-designs of order \(v\) intersecting in \(s\) blocks and \(2s+t\) triangles\(\}\). Let \(\mathrm{Adm}(v)=\{(s,t):s+t\leq b_{v},s\in J(v),2s+t\in J_{T}(v)\}\setminus \{(b_{v}-3,1)\}\), where \(J(v)\) (or \(J_{T}(v)\)) denotes the set of positive integers \(s\) (or \(t\)) such that there exists a pair of \((K4-e)\)-designs of order \(v\) intersecting in \(s\) blocks (or \(t\) triangles), and \(b_{v}=v(v-1)/10\). It is established that \(\mathrm{Fin}(v)=\mathrm{Adm}(v)\) for any integer \(v\equiv 0,1\pmod 5\), \(v\geq 6\) and \(v\neq 10,11\).
MSC:
| 05B05 | Combinatorial aspects of block designs |
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