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Statement

Theorem 5.2 (p. 8). If every hyperplane of $\operatorname{AG}(n,q)$ contains at least $t$ points of a point set $A$, then

$$|A|\ge(n+t-1)(q-1)+1.$$

The paper attributes the theorem to A. A. Bruen (J. Combin. Theory Ser. A 60 (1992), 19–33), the case $t=1$ to R. Jamison (J. Combin. Theory Ser. A 22 (1977), 253–266), with an independent proof by A. E. Brouwer and A. Schrijver (J. Combin. Theory Ser. A 24 (1978), 251–253). It notes (p. 8) that S. Ball (European J. Combin. 21 (2000), 441–446) improves the bound slightly in many cases when $t\le q$.

Proof pointer

P. 8. Take $n$ lines through a point $x$ of $A$ spanning $\operatorname{PG}(n,q)$, the hyperplane at infinity $H$, which contains no point of $A$, $S_i=l_i\setminus\{x\}$ and $D_i=l_i\cap H$. Theorem 5.1, applied to the points of $A$ other than $x$, gives $|A|-1\ge(t-1)(q-1)+n(q-1)$.

Read depth

Claims checked: the statement was read clause by clause against p. 8 of the print, and the proof was followed.

Dependencies

Theorem 5.1.

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.