Linear embeddings of random complexes

For $X\sim X(n;1,n^{-{\alpha }_1},n^{-{\alpha }_2},\mathrm{.}\mathrm{.}\mathrm{.})$ in the multiparameter random simplicial complex model we establish necessary and sufficient strict inequalities on the ${\alpha }_i$’s to linearly embed the complex into ${\mathbb{R}}^{2d}$.