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Source. Section 6.1, p. 8, 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. $q(r)$ is the minimum number of edges of an intersecting $r$-uniform hypergraph $H$ with covering number $\tau(H)=r$; see Theorem 6.7 for the terms. The paper credits Tripathi with showing that the Erdős-Lovász bound $\lceil\frac83\cdot4-3\rceil=8$ is not attained and with a 9-edge example, so $q(4)=9$ (pp. 2, 8).

Result (p. 8, unnumbered). Up to isomorphism there is exactly one 4-uniform intersecting hypergraph with 9 edges and covering number 4. It has 11 vertices, and the paper prints its incidence matrix (p. 8). In the paper's words, "This is a unique example with 9 edges."

The introduction (p. 2) and the abstract (p. 1) add that this example "is not symmetric by any means"; the paper gives no further argument for that remark.

Read depth. Claims checked: the statement and the counting step were read on the print; the computer search was not rerun. Nothing here is independently reviewed.

Proof pointer

p. 8. Fewer than 9 vertices is ruled out by the handshake lemma. For 9 to 13 vertices the paper reports an exhaustive search over intersecting 4-uniform hypergraphs with 9 edges satisfying the necessary degree conditions; only one candidate, on 11 vertices, has covering number 4 (table, p. 8). For 14 or more vertices, double counting the 72 ordered intersecting pairs of edges against the vertex degrees, which lie between 2 and 4 with at most one vertex of degree 4, gives fewer than 72 pairs.

Dependencies

Within the paper: Observations 6.1 and 6.2 (pp. 7--8) and the computer search reported on p. 8.

Bears on

• Problem 21: the problem's $f(n)$ is the paper's $q(n)$. The result describes the extremal family for $f(4)=9$; it says nothing about the growth of $f(n)$.