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