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

, 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.