Random Subcomplexes and Betti Numbers of Random Edge Ideals

Abstract We study homological properties of random quadratic monomial ideals in a polynomial ring $R = {\mathbb {K}}[x_1, \dots x_n]$, utilizing methods from the Erd̋s–Rényi model graphs. Here, for graph $G \sim G(n, p),$ we consider coedge ideal $I_G$ generated by monomials corresponding to missing edges $G$ and Betti numbers $R/I_G$ as $n$ tends infinity. Our main results involve setting edge probability $p p(n)$ so that asymptotically almost surely Krull dimension is fixed. Under these conditions, establish various regarding table $R/I_G$, including sharp bounds on regularity projective distribution nonzero normalized numbers. These extend work Erman Yang who studied such context conjectured phenomena nonvanishing asymptotic syzygies. Along way, subcomplexes clique complexes well notions higher-dimensional vertex $k$-connectivity may be independent interest.