Trivial finite projective planes
Task number: 3892
We replace the axiom of finite projective planes on the existence of four points in the general position by the fact that each line contains at least two points. Each final projective plane according to the original definition will satisfy the new one, but the opposite is not true. What other set systems does the new definition allow?
Solution
When \(X=\{1,…,n\}\), the altered axioms will also hold for:
- the system consisting of a single line: \(\mathcal P=\{X\}\)
- system consisting of one line with \(n-1\) points and \(n-1\) two-point lines: \(\mathcal P=\{\{1,…,n-1\},\{i,n: i\ne n\}\}\)