---

Statement

Theorem 5.4 (p. 9). Let $h_1,\ldots,h_n$ be positive integers and $G$ the set of points $(y_1,\ldots,y_n)$ with $y_i\in\mathbb Z$ and $0\le y_i\le h_i$. A set of hyperplanes of $\operatorname{AG}(n,\mathbb R)$ that covers all but one point of $G$ has at least $h_1+h_2+\cdots+h_n$ members.

The phrase "all but one point" means that exactly one point of $G$ lies on no hyperplane of the set, as in the case $t=1$ of Theorem 5.3. The paper attributes the theorem to N. Alon and Z. Füredi, Covering the cube by affine hyperplanes, European J. Combin. 14 (1993), 79–83.

Proof pointer

P. 8: the paper calls it an immediate corollary of Theorem 5.3 with $t=1$; take $S_i=\{0,1,\ldots,h_i\}$ and $D_i$ the coordinate of the uncovered point.

Read depth

Claims checked: the statement was read clause by clause against p. 9 of the print.

Dependencies

Theorem 5.3.

Source. Simeon Ball and Oriol Serra, Punctured combinatorial Nullstellensätze, Combinatorica 29 (2009), 511–522, doi:10.1007/s00493-009-2509-z. Labels and page numbers are those of the corrected author manuscript dated 14 June 2011, the edition named on the source card.

Bears on

None recorded. The paper names no Erdős problem.