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Statement

Setting (p. 133). Breaking up a line of a 2-design means replacing it by the lines of some 2-design on the same set of points; a near pencil on $m$ points is the 2-design consisting of one line of $m-1$ points and the $m-1$ pairs joining the remaining point to them. See Theorem 1 for the definition of a 2-design.

Theorem 3 (p. 133, quoted). "If $v=p^2+p+1$ and $b=p^2+2p+1$, then the design is obtained from a projective plane of order $p$ by "breaking up" one of its lines into a near pencil or projective plane."

The paper calls Theorem 3 sharp in some sense and then proves the stronger Theorem 4 (p. 133).

Proof pointer

The paper gives two proofs. The algebraic proof (p. 138) splits the points into those of degree $p+1$ and those of larger degree, call the latter set $Z$, and applies the projection-matrix argument of the algebraic proof of Theorem 2 to a set of lines indexed by $Z$ to get $|Z|=p+1$; counting pairs then shows that the short lines form a possibly degenerate projective plane on $Z$, and the long lines together with $Z$ form a projective plane of order $p$. The combinatorial proof (pp. 138--141) is that of Theorem 4, whose equality case is that exactly one line of a projective plane of order $p$ was broken up. The paper notes (p. 133) that Theorem 3 also follows from Totten's classification of the 2-designs with $(b-v)^2\le v$, by a longer proof.

Read depth

Claims checked: Theorem 3 and the definitions it uses were read clause by clause on the page images of the print, and the algebraic proof on p. 138 was followed for structure. Nothing here is independently reviewed.

Dependencies

Lemmas 1 to 4 (pp. 135--136) and the algebraic proof of Theorem 2. External inputs named by the paper: the de Bruijn--Erdős theorem, and Vanstone's embedding theorem in the combinatorial proof.

Source. P. Erdős, J. C. Fowler, V. T. Sós and R. M. Wilson, On 2-designs, J. Combin. Theory Ser. A 38 (1985), no. 2, 131--142; the edition read is named on the source card.

Bears on

• Problem 903: Theorem 3 describes every design attaining the bound $n+p$ of the problem; it does not bear on the problem's question beyond that.