Let v=19. How many blocks does an STS(19) have, how many blocks pass through each point, and does v≡ 1,3 pmod 6 hold?

b=57, r=18, and it does not hold

b=57, r=9, and it does not hold

Exercise · ZW-STEINER-TRIPLE-SYSTEM-010 · ★★★ Advanced

Let \(v=19\). How many blocks does an \(\mathrm{STS}(19)\) have, how many blocks pass through each point, and does \(v\equiv 1,3 \pmod 6\) hold?

Choose one option.

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A. \(b=57\), \(r=9\), and yes it holds

Explanation

With \(b=\frac{19\cdot 18}{6}=57\) and \(r=\frac{18}{2}=9\), and \(19\equiv 1\pmod 6\), the condition holds. Answering \(r=18\) uses \(v-1\) instead of \((v-1)/2\).

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Review the concept: Steiner triple system

A Steiner triple system \(\mathrm{STS}(v)\) is a block design in which \(v\) points are grouped into triples so that every pair of distinct points lies in exactly one triple.

\[b=\frac{v(v-1)}{6},\quad r=\frac{v-1}{2}\]

Let \(v=19\). How many blocks does an \(\mathrm{STS}(19)\) have, how many blocks pass through each point, and does \(v\equiv 1,3 \pmod 6\) hold?

Explanation

Review the concept: Steiner triple system

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