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Source. Section 2, p. 3, and Section 5, p. 7, of J. Barát, "Intersecting and 2-intersecting hypergraphs with maximal covering number: the Erdős-Lovász theme revisited," J. Combin. Des. 29 (2021), no. 3, 193--209. The edition read, and whose pages are cited, is identified on the source card. The paper gives these findings no label.

Statement

Setting (p. 3). A biplane is a symmetric 2-design with $\lambda=2$: every two points lie in exactly two blocks and every two blocks meet in exactly two points. A biplane of order $n$ has blocks of size $k=n+2$, so its blocks form a $2$-intersecting $k$-uniform hypergraph, whose covering number is at most $k-1$; it is maximal when it equals $k-1$ (see Proposition 3.1 for the terms). The paper cites 18 known biplanes.

Result (unlabelled). The paper reports the following covering numbers.

• Order 1 ($k=3$, the tetrahedron): $\tau=2$, maximal (p. 3).
• Order 2 ($k=4$, the complement of the Fano plane): $\tau=3$, maximal (p. 3).
• Order 3 ($k=5$, the Paley biplane on 11 points): $\tau=4$, maximal, checked by computer (p. 3).
• Order 4 ($k=6$), all three biplanes: $\tau=4$, checked by computer; two of them were first checked by hand, and the paper gives the hand argument for one, the Kummer configuration (p. 3).
• Order 7 ($k=9$), the four biplanes B9A, its dual, B9B and B9C of Royle's list: each has a 7-cover, printed (p. 7).
• Order 9 ($k=11$), the five biplanes B11A to B11E: each has a 9-cover, printed (p. 7).
• Order 11 ($k=13$), B13A, called the only known one: an 11-cover, printed (p. 7).

The paper concludes on p. 7 that the known biplanes of orders 7, 9 and 11 do not have maximal covering number, and the abstract (p. 1) states "We determined that only 3 biplanes of the 18 known examples are extremal." The lists above name 16 biplanes; the paper does not identify the other two of the 18.

Read depth. Claims checked: the reported covering numbers and covers were read on the print; the computer checks were not rerun and the printed covers were not checked against Royle's lists. Nothing here is independently reviewed.

Proof pointer

pp. 3 and 7. Each non-maximal case is witnessed by an explicit cover of size $k-2$ or less; the maximal cases and the order-4 values come from an exhaustive cover search by computer, and by hand for the tetrahedron, the complement of the Fano plane and the Kummer configuration.

Dependencies

The lists of biplanes on G. Royle's home page on biplanes (ref. [15]) and the incidence matrices in the DistanceRegular.org database (ref. [4]), which the paper cites; the count of 18 known biplanes is cited from Wikipedia on biplanes (ref. [17]).

Bears on

No Erdős problem is linked to this result. It bears on the paper's own Problem 2.1 (p. 3) through the biplanes the paper names as a geometric source of examples other than $\binom{2r-2}{r}$.