Axiomatization only by cardinality

Task number: 3897

Let \((X,\mathcal P)\) be a set system and for \(n \in \mathbb N\), \(n \geq 2\), it holds that:

  1. \(|X|=|\mathcal P|=n^2+n+1\),
  2. \(\forall P \in \mathcal P: |P|=n+1\) a
  3. \(\forall x\in X: |\{P \in \mathcal P: x \in P\}|=n+1\).

Is then \((X,\mathcal P)\) a finite projective plane?

  • Solution

    No, there are counterexamples of set systems that meet these conditions, but they are not projective planes.

    E.g. in general for \( a = n + 1 \) and \( b = n ^ 2 + n + 1 \), points can be chosen \(X=\{x_0, x_1, \ldots, x_{b-1}\}\) and ‘lines’ \(P_i=\{x_i, x_{i+1}, \ldots, x_{(i+a-1)\bmod b}\}\) for \(i=0{,}1,\ldots,b-1\). With this construction we get a system with \(b\) points and with \(b\) ‘lines’.

    Each of the ‘lines’ has a power of \(a\) and each point is in \(a\) ‘lines’. However, the pair of points \(x_i, x_{i+1} \) is in \( a-1 \) common ‘lines’, and therefore it is not a projective plane.

Difficulty level: Easy task (using definitions and simple reasoning)
Proving or derivation task
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