---

Statement

Setting. $n(a,b)$ and $n_1(a,b)$ are as defined on the Lemma 4 page (p. 135); $[x]$ is the integer part.

Proposition 7 (p. 139). If $a\ge1$, then $n(a,0)=n_1(a,0)=a$ and $n(a,1)=n_1(a,1)=[((a+2)/2)^2]$.

Through Theorem 19 the paper derives from it (p. 143) a corollary of Lehel and the Erdős--Gallai bound: an $r$-uniform $\tau$-critical hypergraph with $\tau=2$ has at most $[((r+2)/2)^2]$ vertices.

Source. Zs. Tuza, Critical hypergraphs and intersecting set-pair systems, J. Combin. Theory Ser. B 39 (1985), no. 2, 134--145, doi:10.1016/0095-8956(85)90043-7, as identified on the source card: Proposition 7 on p. 139.

Read depth. Claims checked: the statement was read on the print and the proof on p. 139 was followed. Nothing here is independently reviewed.

Proof pointer

Page 139. For $b=0$ every $B_i$ is empty, so there is one pair. For $b=1$, with $m$ pairs, the points $B_1,\ldots,B_m$ are distinct and each $A_i$ contains exactly $m-1$ of them, so the union has at most $m(a-m+1)+m=m(a+2-m)\le[((a+2)/2)^2]$ points. The proof prints only this upper bound. The reverse inequality is a check made here, not in the paper: the paper's Construction 1 (p. 136) with $b=1$ and $a'=\lceil a/2\rceil\ge1$ has $a'+1\ge2$ pairs, whose first coordinates cover all $a'+1+(a-a')(a'+1)=(a'+1)(a+1-a')=[((a+2)/2)^2]$ points.

Dependencies

None in the corpus.

Bears on

No problem page uses the proposition directly.