It is useful to know the following Erdos-Szekeres fact: For every k > d there is N(k,d) so that every N points in general position in R^d contains k points in "cyclic position". We say that d points $x_1, x_2,...x_d$ are in cyclic position if all the simplices $x_{i_1},...,x_{i_{d+1}}$ have the same orientation. This fact follows from Ramsey's theorem. (With very large N(k,d).) It implies various results of the kind ask here if they refer to properties of points in cyclic positions.
This gives a complete answer for the case $k=2$ of the original question. Indeed it is useful to think about the original problem as for which sizes $d_1,d_2\dots,d_k$ whose sum is $(d+1)(k-1)+1$ if N is large enough and we have $N$ points in $R^d$ we can find a subset of $(d+1)(k-1)+1$ points with Tverberg partition of sizes $d_1,d_2,\dots,d_k$. When it comes to Radon partitions points in cyclic position are "cannonical". For larger values of $k$ I dont know the precise situation. We can look at lexicographic sequence of points on the moment curve and this is a property inherited by subsequences. (So it will exclude plenty of $d_i$ sequences.) But I am not sure every large set of points "contains" a lexicographic sequence of points on the moment curve. ("contains" in terms of having equivalent Tverberg's behavior.) So there is more to explore.
By the way, an interesting higher dimensional question is what is the number f(n,d) so that every f(n,d) points in general position in $R^d$ contains $n$ points in convex position. This is monotonic non-increasing in $d$.
It is useful to know the following Erdos-Szekeres fact: For every k > d there is N(k,d) so that every N points in general position in R^d contains k points in "cyclic position". We say that d points $x_1, x_2,...x_d$ are in cyclic position if all the simplices $x_{i_1},...,x_{i_{d+1}}$ have the same orientation. This fact follows from Ramsey's theorem. (With very large N(k,d).) It implies various results of the kind ask here if they refer to properties of points in cyclic positions.