Here we use standard notation for Tverberg's theorem: Dimension $d$, number of partition blocks $r$, and $N=(r-1)(d+1)$.
The configuration space of Tverberg's theorem is the simplicial complex $K_1=(\Delta_N)^{*r}_{\Delta(2)}$. It has $r\times (N+1)$ vertices which are often shown as a grid of $r$ columns, and $N+1$ rows. The symmetric group on $r$ elements acts on it by permuting its columns.
One of the key steps of Vucic & Zivaljevic (1993), Hell (2007) for proving lower bounds on the number of Tverberg partitions is their construction of a large family of subcomplexes of $K_1$ each containing a maximal face corresponding to an ordered (!) Tverberg partition. The subcomplexes are defined by choosing two points from every row of $K_1$. Joining two points from every row leads to a $N$-dimensional sphere in $K_1$.
In 2009, Blagojevic, Matschke & Ziegler used the simplicial complex $$K_2= \Delta_{r,C_1} * \Delta_{r,C_2} *\cdots * \Delta_{r,C_m}$$ to obtain their optimal colorful Tverberg theorem. Here $K_2$ is the join of $m$ chessboard complexes $\Delta_{r,C_i}$, and $\Delta_{r,C_i}=(C_i)^{*r}_{\Delta(2)}$ is the chessboard complex on $r$ columns and $|C_i|$ rows. $K_2$ is a subcomplex of $K_1$ carrying the action from above.
As far as I understand, choosing two points from every row does again lead to $N$-dimensional sphere, but this is in general not in $K_2$. If one of the color classes has $l>\frac{r}{2}$ points, the corresponding chessboard complex has $r$ columns, and $l$ rows. Therefore two of the chosen points are from the same column. This is not allowed in the chessboard complex.
Question 1: Is there anything known when all color classes in the setting of the optimal colorful Tverberg theorem have at most $\frac{r}{2}$ elments?
I'm looking for a construction: Given a maximal face corresponding to an ordered (!) colored Tverberg partition. Choose a second point from every row that
- leads to a subcomplex $L$ of $K_2$
- which meets the orbit of the maximal face only once (in other words: contains the unordered (!) colored Tverberg partition only once.)
Question 2: Does such a subcomplex always exist? Can the second point always be chosen this way?
Any comments on this are very welcome. (In fact, there is a also third condition on $L$. If interested I'm ready to share. Here it would make the post to long.)