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Source. Proposition 3.1, p. 6 (Section 3, pp. 5--6), 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.

Statement

Setting (pp. 2--3). A hypergraph is $2$-intersecting when any two edges share at least two vertices. In a $2$-intersecting $r$-uniform hypergraph any edge with one vertex removed still meets every edge, so $\tau\le r-1$; the paper studies those with $\tau=r-1$, its maximal covering number in this setting. Its standard example is $\binom{2r-2}{r}$, all $r$-subsets of a $(2r-2)$-element set. The hypergraphs considered are simple: no edge is repeated (p. 1).

Proposition 3.1 (p. 6). "There are precisely two non-isomorphic 4-uniform 2-intersecting hypergraphs that have covering number 3, namely $\binom{6}{4}$ and the complement of the Fano plane."

The complement of the Fano plane, the 7 complements of the lines of the Fano plane, is the biplane of order 2 (p. 3). The $3$-uniform analogue is Proposition 2.7 (p. 4): the only $3$-uniform $2$-intersecting hypergraph with maximum covering number is $\binom{4}{3}$, the biplane of order 1.

Read depth. Claims checked: the statement was read on the print and the case analysis followed in outline. Nothing here is independently reviewed.

Proof pointer

pp. 5--6, a case analysis by hand on the incidence matrix. Since no two vertices cover, every pair of rows has a column with zeros in both; this fixes the first four rows and seven columns up to isomorphism. Requiring every pair of the remaining edges to share two vertices leaves few completions: one gives $\binom{6}{4}$, one gives the complement of the Fano plane, and each of the others either forces an edge meeting another in one vertex or keeps a $2$-cover that no admissible new edge can avoid.

Dependencies

None outside Section 3.

Bears on

No Erdős problem is linked to this result. It bears on the paper's own Problem 2.1 (p. 3), which asks whether there are infinitely many $r$ for which $\binom{2r-2}{r}$ is the only $2$-intersecting $r$-uniform hypergraph with maximal covering number, and whether there are infinitely many other examples. The proposition settles only the case $r=4$, where $\binom{6}{4}$ is not the only example; it decides neither question.