There is no mni clutter whose core is the projective plane of order 3

In 1998, Lütolf and Margot claimed, in a sidenote, that the projective plane of order 3 3 , also called the ternary plane, or PG(2,3 P G ( 2 , 3 ), cannot be the core of any minimally non-ideal clutter. In this paper, we show that, if \operatorname{\cal C} C ⁡ is a minimally non-ideal clutter whose core is the PG(2,3) P G ( 2 , 3 ) then \operatorname{\cal C} C ⁡ must be the blocker of the PG(2,3) P G ( 2 , 3 ) . Since no projective plane of order n \geq 3 n ≥ 3 is minimally non-ideal, the nonexistence of \operatorname{\cal C} C ⁡ is a consequence of the fact that a clutter is minimally non-ideal if and only its blocker is minimally non-ideal.