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Statement

Theorem 5.5 (p. 9). Let $S_1,\ldots,S_n$ be finite nonempty subsets of a field $\mathbb F$. If $m$ hyperplanes of $\operatorname{AG}(n,\mathbb F)$ do not cover $S_1\times\cdots\times S_n$, then they leave uncovered at least $\min\prod_{i=1}^n y_i$ of its points, the minimum taken over positive integers $y_i\le|S_i|$ with $\sum_{i=1}^n y_i\ge\sum_{i=1}^n|S_i|-m$.

The print bounds $y_i$ by $|S_n|$; this is read as $|S_i|$, the bound in Corollary 4.3, from which the paper says the theorem follows directly. The paper identifies the theorem with Theorem 4 of N. Alon and Z. Füredi, Covering the cube by affine hyperplanes, European J. Combin. 14 (1993), 79–83.

Proof pointer

P. 9: apply Corollary 4.3 to the product of the $m$ affine linear forms defining the hyperplanes, a polynomial of degree $m$ that is nonzero exactly at the uncovered points.

Read depth

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

Dependencies

Corollary 4.3.

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.