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Statement

Setting (p. 6). $\mathbb F$ is an arbitrary field and $\operatorname{PG}(n,\mathbb F)$ the $n$-dimensional projective geometry over it.

Theorem 5.1 (p. 7). Let $t$ be a positive integer and $l_1,\ldots,l_n$ concurrent lines through a point $x$ that span $\operatorname{PG}(n,\mathbb F)$. Let $S_i$ be a set of points of $l_i\setminus\{x\}$ and $D_i$ a proper nonempty subset of $S_i$. Suppose $A$ is a set of points such that every hyperplane $\langle s_1,\ldots,s_n\rangle$ with $(s_1,\ldots,s_n)\in(S_1\times\cdots\times S_n)\setminus(D_1\times\cdots\times D_n)$ contains at least $t$ points of $A$. If some hyperplane $\langle d_1,\ldots,d_n\rangle$ with $(d_1,\ldots,d_n)\in D_1\times\cdots\times D_n$ contains no point of $A$, then

$$|A|\ge(t-1)\max_j\bigl(|S_j|-|D_j|\bigr)+\sum_{i=1}^n\bigl(|S_i|-|D_i|\bigr).$$

The sets $S_i$ are finite, as in the introduction (p. 1). The paper adds (pp. 7–8) that the proof gives the same bound for a multiset $A$, that the hypothesis of a hyperplane missing $A$ cannot be dropped, and that for $t=1$ the bound is attained by $A=\bigcup_{i=1}^n(S_i\setminus D_i)$. The introduction (pp. 1–2) illustrates the case $n=2$, $t=1$, where the bound is $|S_1|+|S_2|-2$ when $D_1$ and $D_2$ are single points.

Proof pointer

P. 7. A collineation sends a hyperplane through points of the $D_i$ that misses $A$ to the hyperplane at infinity and the lines $l_i$ to the coordinate axes. The hyperplanes spanned by points of the $S_i$ then become $\sum_i t_iX_i=1$ with $t_i$ in parameter sets $T_i$, and $f=\prod_{a\in A}\bigl(\sum_i a_iX_i-1\bigr)$ has a zero of multiplicity $t$ outside the smaller parameter grid and is nonzero at the origin. Theorem 4.1 bounds $\deg f$. The print writes $|A|=\deg f$; when $x\in A$ the factor of $x$ is constant, and the argument uses only $\deg f\le|A|$.

Read depth

Claims checked: the statement and the remarks after it were read clause by clause against pp. 7–8 of the print, and the proof was followed.

Dependencies

Theorem 4.1.

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.