We prove a uniform bound on the topological Turán number of an arbitrary two-dimensional simplicial complex S: any n vertices with at least CSn3−1/5 facets contains homeomorph S, where CS > 0 is constant depending S alone. This result, analogue classical one-dimensonal result Mader, sheds some light old problem Linial from 2006.

Let $\mathbf L_k$ be the holomorphic line bundle of degree $k \in \mathbb Z$ on projective line. Here, tuples $(k_1 k_2 k_3 k_4)$ for which there does not exists homogeneous non-split supermanifolds $CP^{1|4}_{k_1 k_4}$ associated with vector L_{−k_1} \oplus \mathbf L _{−k_2} L_{−k_3} L_{−k_4}$ are classified. \\For many types remaining tuples, listed cocycles that determine supermanifolds. \\P...

in this paper, it is proved that all simple $k_4$-groups of type $l_2(q)$ can be characterized by their maximum element orders together with their orders. furthermore, the automorphism groups of simple $k_4$-groups of type $l_2(q)$ are also considered.

A K4-homeomorph is a subdivision of the complete graph with four vertices (K4). Such a homeomorph is denoted by K4(a,b,c,d,e,f) if the six edges of K4 are replaced by the six paths of length a,b,c,d,e,f, respectively. In this paper, we discuss the chromaticity of a family of K4-homeomorphs with girth 10. We also give sufficient and necessary condition for some graphs in the family to be chromat...