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Statement

Setting (p. 131). A 2-design on $v$ points is a family of lines (subsets with more than one point) such that every pair of points lies in exactly one line; see Theorem 1 for the full setting.

Theorem 2 (p. 132, quoted). "Let $v=p^2+p+1$. Then for $p^2+p+1<b<p^2+2p+1$ there is no 2-design with $v$ points and $b$ lines."

Here $p$ is a positive integer, not necessarily a prime or a prime power (p. 132). In the abstract's terms (p. 131), the interval $[v+1,v+p-1]$ is disjoint from $M_v$.

Sharpness (p. 132). The bound is best possible whenever a projective plane of order $p$ exists: in such a plane replace one line $\{x_1,\ldots,x_{p+1}\}$ by the line $\{x_2,\ldots,x_{p+1}\}$ and the $p$ pairs $\{x_1,x_i\}$, $2\le i\le p+1$. The result is a 2-design with $p^2+2p+1$ lines; the old line has been broken up into a near pencil on its $p+1$ points (p. 133).

Remark (p. 132). The result fails for $v$ not of this form: projective planes from which points have been deleted give many examples with $b-v<\sqrt v$.

Proof pointer

The paper gives two proofs. The algebraic proof (pp. 135--137) first shows that a line of more than $p+1$ points forces a near pencil (Lemma 1, p. 135), so that all lines have at most $p+1$ points and every point has degree at least $p+1$ (Lemma 3, p. 136), and that a line all of whose points have degree $p+1$ forces $b=v$ when $b\le p^2+2p+1$ (Lemma 2, p. 136). It then studies the orthogonal projection onto the complement of the row space of the $v\times b$ incidence matrix, which has rank $b-v$, and a principal submatrix indexed by the lines through a point $x_0$ of degree $p+1$ (which exists by Lemma 4, p. 136); if $b-v\le p-1$, a positivity argument forces a line through $x_0$ all of whose other points have degree $p+1$, and Lemma 2 finishes. The combinatorial proof (pp. 138--141) derives Theorem 2 from Theorem 4, since breaking up a line of a projective plane gives at least $p^2+2p+1$ lines by the de Bruijn--Erdős theorem.

The paper also notes (p. 133) that Theorems 2 and 3 follow from Totten's classification of the 2-designs with $(b-v)^2\le v$, by a substantially longer proof.

Read depth

Claims checked: Theorem 2, the sharpness construction and the remark were read clause by clause on the page images of the print, and the algebraic proof with Lemmas 1 to 4 was followed. Nothing here is independently reviewed.

Dependencies

Theorem 4 for the combinatorial proof. External inputs named by the paper: the de Bruijn--Erdős theorem and the Stanton--Kalbfleisch bound used in Lemma 1.

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 2 is the problem's assertion, with blocks of at least two points: a 2-design on $n=p^2+p+1$ points with more than $n$ blocks has at least $n+p$. The theorem needs no prime-power hypothesis on $p$, and the construction above attains $n+p$ when $p$ is a prime power.