Tverberg Plus Minus

NEW

Problem: Reconstructed statement: For integers $d\ge1,r\ge2$, let $F_1,\dots,F_{d+1}\subset\mathbb R^d$ be pairwise disjoint color classes of size $r$, and let $M\subset[d+1]$. Does there always exist a colorful partition $A_1,\dots,A_r$, $|A_j\cap F_i|=1$, and a common point $z\in\mathbb R^d$ with affine representations

$$z=\sum_{i=1}^{d+1}\alpha_{i,j}x_{i,j},\qquad \sum_{i=1}^{d+1}\alpha_{i,j}=1,$$

where $x_{i,j}=A_j\cap F_i$, such that either all coefficients with $i\in M$ are negative and all coefficients with $i\notin M$ are positive, or the reverse sign pattern holds? This is Theorem 4.1 of Bárány–Soberón with the equal-coefficients condition removed and $n=d+1$.

Result: The statement is false already for $d=1,r=3$.

Choose two color classes

$$F_1=\{a_1,a_2,a_3\},\qquad F_2=\{b_1,b_2,b_3\}\subset\mathbb R$$

with algebraically independent coordinates and order

$$a_1<b_1<b_2<a_2<a_3<b_3.$$

Let $M=\{1\}$.

In one dimension, for a pair $(a,b)$ with coefficient of $a$ negative and coefficient of $b$ positive, the common point $z$ must lie beyond $b$ on the ray from $a$ through $b$; equivalently, $b$ lies strictly between $a$ and $z$.

Suppose first $F_1$ has negative coefficients and $F_2$ positive coefficients. Then $b$-points must be matched to $a$-points so that each matched $b$ lies between its $a$ and the same $z$.

If $z>a_3$, then both $a_2$ and $a_3$ require a matched $b$-point to their right, but the only such $b$-point is $b_3$, impossible. Hence $z<a_3$. Then $a_2,a_3$ must both be matched to distinct points among $b_1,b_2$, forcing $z<b_1$. But then $a_1$ has no $b$-point between $a_1$ and $z$, contradiction.

The reverse sign pattern is the same argument after interchanging the two colors and applying $x\mapsto -x$, since the transformed order is again

$$n_1<p_1<p_2<n_2<n_3<p_3.$$

Thus neither sign alternative is possible for any colorful partition.

So Theorem 4.1 cannot hold with $n=d+1$ after removing the equal-coefficients condition.

Citation: Counterexample above. The reconstructed problem is from Bárány–Soberón, “Tverberg Plus Minus,” Open Problem 4.1/4.2.