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[x1, ..., x(n)], generated by all squarefree monomials of degree 3 in the variables x1, ..., 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.