Revisiting the spreading and covering numbers (Q2848724)

Let \(\mathbb{K}[x_1,\dots,x_n]\) be a polynomial ring over an arbitrary field \(\mathbb{K}\) and let \(M_d\) be the set of all monomials of degree \(d\) in the \(\mathbb{K}[x_1,\dots,x_n]\), where \(d\) is a non-negative integer. We denote by \(M_1W=\{x_im : m\in W \text{ and } 1\leq i\leq n\}\) a monomial subspace of \(M_{d+1}\). Correspondingly, the spreading number NEWLINE\[NEWLINE\alpha_n(d)=\max \{\dim_k(W) : W\subseteq M_d \text{ and } \dim_k(M_1W)=n\dim_k(W)\}NEWLINE\]NEWLINE and the covering number NEWLINE\[NEWLINE\rho_n(d+1)=\min \{\dim_k(W) : W\subseteq M_d \text{ and } M_1W=M_{d+1}\}NEWLINE\]NEWLINE are defined. These numbers had been introduced by Geramita et al in order to study Cohen-Macaulay and ideal generation conjectures, see [\textit{A. V. Geramita} et al., J. Pure Appl. Algebra 40, 33--62 (1986; Zbl 0586.13015)]. Also by \(S_n(d)\) we denote the graph on the vertex set \(V=\{m_i\in M_d\}\) and on the edge set \(E=\{(m_i,m_j) : \deg(\mathrm{lcm}(m_i,m_j))=d+1\}\).NEWLINENEWLINEThe authors prove the following result in their article:NEWLINENEWLINEFor all \(d\geq0\), \(\alpha_4(d)\) equals the number of non-negative integer \(2\times2\) matrices with sum of entries equal to \(d\), under row and column permutations.NEWLINENEWLINEIn order to prove this, they are using combinatorics on the OEIS (On-line Encyclopedia of Integer Sequences). Finally, to compute improved upper bounds for the covering number \(\rho_n(d)\), they give an algorithm which is based on the construction and the study of minimal clique cover of the graph \(S_n(d)\).