Let \(S \subseteq \mathbb{N}^n\) be a semigroup. Recall that \(S\) is called a homogeneous semigroup if S \(S=\cup_{i\geq 0}S_i\) is a disjoint union, where \(S_0 = \{0\}\), \(S_i + S_j \subseteq S_{i+j}\), for all \(i, j\) and \(S\) is generated by some elements \(\mathbf{s}_1, \ldots, \mathbf{s}_m\) of \(S_1\). The elements of \(S_i\) are called elements of degree \(i\). Let \(\mathbb{K}\) be a field and \(\mathbf{x}^{\alpha}=x_1^{\alpha_1} \cdots x_n^{\alpha_n}\) be a monomial in \(R=\mathbb{K}[x_1, \ldots, x_n]\) where \(\alpha=(\alpha_1, \ldots, \alpha_n) \in \mathbb{N}^n\). Let \(A = \mathbb{K}[S]\) be the \(\mathbb{K}\)-subalgebra of \(R\) generated by monomials \(\mathbf{x}^{\mathbf{s}_1}, \ldots, \mathbf{x}^{\mathbf{s}_m}\) in \(R\). It is clear that the semigroup ring \(\mathbb{K}[S]\) of a homogeneous semigroup \(S\) is a homogeneous \(\mathbb{K}\)-algebra with homogeneous components \(\mathbb{K}[S]_i=\mathbb{K}S_i\). Following [J. Herzog et al., Math. Scand. 86, No. 2, 161–178 (2000; Zbl 1061.13008)], an standard graded \(\mathbb{K}\)-algebra is called strongly Koszul if its graded maximal ideal \(\mathfrak{m}\) admits a minimal system of homogeneous generators \(u_1, \ldots, u_m\), such that for all subsequences \(u_{i_1}, \ldots, u_{i_r}\) of \(u_1, \ldots, u_m\), with \(1 \leq i_1 < i_2 < \cdots < i_r \leq m\) and for all \(j = 1, \ldots, r\) the colon ideal \((u_{i_1}, \ldots, u_{i_{j-1}}) \colon u_{i_j}\) is generated by a subset of \(\{u_1, \ldots, u_m\}\). Also a homogeneous semigroup \(S\) is called strongly Koszul if \(\mathbb{K}[S]\) is strongly Koszul. This property only depends on \(S\) but not on the field \(\mathbb{K}\). The intersection degree \(b(S)\) of a semigroup \(S\) is the maximum degree of the generators of the ideals \((u_i)\cap(u_j)\), where \(u_i\) and \(u_j\) are any two generators of the semigroup ring \(\mathbb{K}[S]\). It is known that \(S\) is strongly Koszul if and only if \(b(S)=2\) [J. Herzog et al., Math. Scand. 86, No. 2, 161–178 (2000; Zbl 1061.13008), Proposition 1.4].
Let \(n\) be a positive integer and \([n]=\{1, \ldots, n\}\). For a subset \(F \subseteq [n]\), let \(\mathbf{x}_F=\prod_{i \in F}x_i\) be squarefree monomial in \(R=\mathbb{K}[x_1, \ldots, x_n]\). For integers \(r \leq n\), let \(A^{(r,n)}=\mathbb{K}[\mathbf{x}_F \colon F \subseteq [n] \text{ and } |F|=r]\) be \(\mathbb{K}\)-subalgebra of \(R\) and let \(S(r,n)\) be a semigroup such that \(\mathbb{K}[S(r,n)]=A^{(r,n)}\). It is shown in [A. Aramova et al., Nagoya Math. J. 168, 65–84 (2002; Zbl 1041.16018)] that \[ b(S(2,n))=\begin{cases} 2, & \text{ if } n=2,3\\ 3, & \text{ if } n \geq 4 \end{cases} \] One of the main results of the paper under review is that: \[ b(S(3,n))=\begin{cases} 2, & \text{ if } n=3,4\\ 3, & \text{ if } n =5,6,7\\ 4, & \text{ if } n \geq 8 \end{cases} \] On the other hand, in the paper under review, the author consider the simplicial complex \(\Delta^{(r,n)}=\{ F \subseteq [n] \colon |F| \leq r\}\) and introduce a new property in \(\Delta^{(3,n)}\) for a pure \(2\)-dimensional subcomplex, that is called Triangle-Distance Condition (TDC). The main result is this matter is that:

Theorem. Let \( n\geq 5\) and \(\mathcal{F} = \{\Delta_i\}_{i\in I}\) be the family of all subcomplexes of \(\Delta^{(3,n)}\) such that each each \(\Delta_i\) is pure, connected, chordal and satisfies the TDC. Then, for each \(i\), there exists a facet \(F \in \Delta_i\) such that \(\Delta_i \setminus F \in \mathcal{F}\).

In the above theorem, the concept of chordality for higher dimensional hypergraphs comes from [B. Benedetti and D. Bolognini, J. Comb. Theory, Ser. A 180, Article ID 105430, 10 p. (2021; Zbl 1459.05355); M. Bigdeli et al., J. Comb. Theory, Ser. A 145, 129–149 (2017; Zbl 1355.05285); M. Bigdeli et al., J. Algebra 531, 102–124 (2019; Zbl 1420.13028)] and this theorem generalizes a result of T. Hibi et al. [“A Koszul filtration for the second squarefree Veronese subring”, Int. J. Algebra 9, No. 1, 7–14 (2015)] to simplicial complexes.