On the Third Squarefree Veronese Subring

Let S be a homogeneous semigroup and let A = K [ S ] be the semigroup ring of S on a field K . We investigate on basic properties of the third squarefree Veronese semigroup ring A ( 3 , n ) , subring of K [ x 1 , … , x n ] , generated by all squarefree monomials of degree 3 in the variables x 1 , … , x n , with deg ( x i ) = 1 . We compute the intersection degree b ( A ( 3 , n ) ) and we prove that it is 4, for n ≥ 8 . As an application, we study the simplicial complex on A ( 3 , n ) and we determine filtrations of sets of triangles that guarantee the persistence of good properties from the point of view of the theory of simplicial complexes.