A Tverberg type theorem for collectively unavoidable complexes

We prove (Theorem 2.4) that the symmetrized deleted join $SymmDelJoin(\mathcal{K})$ of a "balanced family" $\mathcal{K} = \langle K_i\rangle_{i=1}^r$ of collectively $r$-unavoidable subcomplexes of $2^{[m]}$ is $(m-r-1)$-connected. As a consequence we obtain a Tverberg-Van Kampen-Flores type result (Theorem 3.2) which is more conceptual and more general then previously known results. Already the case $r=2$ of Theorem 3.2 seems to be new as an extension of the classical Van Kampen-Flores theorem. The main tool used in the paper is R. Forman's discrete Morse theory.