$ \newcommand{\F}{{\mathbb F}}$ Let $ q$ be an odd prime power. A *blocking set* in the affine plane $ \F_q^2$ is a set blocking (meeting) every line.

A union of two non-parallel lines is a blocking set of size $ 2q-1$ , and it is well-known that this is the smallest possible size of a blocking set in $ \F_q^2$ . A very simple proof goes as follows. Suppose that $ B\subset\F_q^2$ is a blocking set. Translating $ B$ appropriately, we can assume that $ 0\in B$ , and we let then $ B_0:=B\setminus\{0\}$ . The new set $ B_0$ blocks every line not passing through the origin; that is, every line of the form $ ax+by=1$ with $ a,b\in\F_q$ not equal to $ 0$ simultaneously. As a result, the polynomial $ $ P(x,y):=\prod_{(a,b)\in B_0}(ax+by-1) $ $ vanishes at every point of $ \F_q^2$ with the exception of the origin. Now, if we had $ |B_0|<2p-2$ , then $ P(x,y)$ would be a linear combination of monomials of the form $ x^my^n$ with $ \min\{m,n\}<p-1$ , while for every such monomial, $ $ \sum_{x,y\in\F_q} x^my^n=0; $ $ this would lead to $ $ \sum_{x,y\in\F_q}P(x,y)=0, $ $ a contradiction.

Suppose now that $ B\subset\F_q^2$ blocks every line

with the possible exception of at most one line in every direction. What is the smallest possible size of such an “almost blocking” set?

If $ B\subset\F_q^2$ is almost blocking, then pairing in an arbitrary way the non-blocked lines and adding to $ B$ the intersection points of these pairs of lines we get a “usual” blocking set; since we had to add at most $ (q+1)/2$ points, this gives $ $ |B| \ge (2q-1)-\frac{q+1}2 = \frac32(q-1). $ $ On the other hand, it is not difficult to construct almost blocking sets $ B\subset \F_q^2$ with $ $ |B| < 2q-\sqrt q. $ $

Is the smallest possible size of an almost blocking set “essentially $ \frac32\,q$ ” or “essentially $ 2q$ ” (or neither)?