Symmetric multiple chessboard complexes and a new theorem of Tverberg type

We prove a new theorem of Tverberg–van Kampen–Flores type, which confirms a conjecture of Blagojevi? et al. about the existence of ‘balanced Tverberg partitions’ (Conjecture 6.6 in Tverberg plus constraints, Bull. London Math. Soc. 46:953–967 (2014]). The conditions in this theorem are somewhat weaker than in the original conjecture, and we show that the theorem is optimal in the sense that the new (weakened) condition is also necessary. Among the consequences is a positive answer (Theorem 7.2) to the ‘balanced case’ of the question asking whether each admissible r-tuple is Tverberg prescribable (Blagojevi? et al. 2014, Question 6.9).