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We study homological properties of random quadratic monomial ideals in apolynomial ring $R = {\mathbb K}[x_1, \dots x_n]$, utilizing methods from theErd\"{o}s-R\'{e}nyi model of random graphs. Here for a graph $G \sim G(n, p)$we consider the `coedge' ideal $I_G$ corresponding to the missing edges of $G$,and study Betti numbers of $R/I_G$ as $n$ tends to infinity. Our main resultsinvolve setting the edge probability $p = p(n)$ so that asymptotically almostsurely the Krull dimension of $R/I_G$ is fixed. Under these conditions weestablish various properties regarding the Betti table of $R/I_G$, includingsharp bounds on regularity and projective dimension, and distribution ofnonzero normalized Betti numbers. These results extend work of Erman and Yang,who studied such ideals in the context of conjectured phenomena in thenonvanishing of asymptotic syzygies. Along the way we establish resultsregarding subcomplexes of random clique complexes as well as notions ofhigher-dimensional vertex $k$-connectivity that may be of independent interest.

Random subcomplexes and Betti numbers of random edge ideals