We prove that the symmetrized deleted join SymmDelJoin( $$\mathcal{K}$$ ) of a “balanced family” = ?Ki? =1 collectively r-unavoidable subcomplexes 2[m] is (m?r?1)-connected. As consequence we obtain Tverberg-Van Kampen-Flores type result which more conceptual and general than previously known results. Already case r 2 this seems to be new as an extension classical Van theorem. The main tool use...

Let `,(n, m) denote the class of simple graphs on n vertices and m edges and let G C `6(n, m) . There are many results in graph theory giving conditions under which G contains certain types of subgraphs, such as cycles of given lengths, complete graphs, etc . For example, Turan's theorem gives a sufficient condition for G to contain a K,,„ in terms of the number of edges in G . In this paper we...