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Statement

Setting (pp. 131--132). A 2-design (pairwise balanced design, linear space) on a finite set $S$ with $|S|=v$ is a family $A_1,\ldots,A_b$ of subsets of $S$ with $|A_i|>1$ for every $i$, such that every pair of elements of $S$ lies in exactly one $A_i$; the $A_i$ are its lines or blocks. $M_v$ is the set of integers $b$ for which a 2-design with $v$ points and $b$ lines exists. The paper records $M_v\subseteq[1,\binom v2]$ and $\binom v2-1,\binom v2-3\notin M_v$, and, by the de Bruijn--Erdős theorem, $b\ge v$ whenever $b>1$. Further, $f(v)$ is the largest integer $b<\binom v2-3$ for which there is no 2-design on $v$ elements with $b$ lines.

Theorem 1 (p. 132, quoted). "There is an absolute constant $c$ so that for $v>v_0$ $f(v)<v+v^{1/2+c}$, where $c$ can be any value $>\frac{11}{40}$."

So, for $v>v_0$, every $b$ with $v+v^{1/2+c}\le b\le\binom v2-4$ lies in $M_v$. The abstract (p. 131) states the consequence that for $v>v_0$, $M_v$ contains the interval $[v+v^{4/5},\binom v2-4]$.

Remark (p. 132). Under plausible assumptions on the distribution of primes the authors say they can prove $f(v)<v+v^{1/2}(\log v)^{\alpha}$ for some fixed $\alpha$, and they conjecture $\limsup_v (f(v)-v)/\sqrt v=\infty$. Near $v$ the situation is different: Theorem 2 excludes every $b$ strictly between $p^2+p+1$ and $p^2+2p+1$ when $v=p^2+p+1$.

Theorem 1* (p. 134). Let $p_k$ be the $k$th prime power in natural order and $v=p_k^2+p_k+1$. Then $f(p_k^2+p_k+1)<p_k^2+2p_k+p_k^{1/2+c}$, where $c$ can be any value $>\frac{11}{40}$.

Proof pointer

Pp. 134--135. The paper proves Theorem 1* and then says that the proof of Theorem 1 can be completed by the same method (p. 135), without giving the details. For Theorem 1*, start from the projective plane of order $p_k$ on the $v$ points and break up its lines into smaller 2-designs to adjust the line count; only $b<p_{k+1}^2+p_{k+1}+1$ needs treating (p. 134). The Heath-Brown--Iwaniec bound $p_{k+1}-p_k<p_k^{11/20+\varepsilon}$ (the paper's reference [5], p. 134, (2)) limits the range of $b$ that must be covered. Erdős's answer to a question of Grünbaum (reference [2], recalled on p. 132: every $b$ with $cv^{3/2}<b\le\binom v2$, other than $\binom v2-1$ and $\binom v2-3$, is the number of lines determined by some $v$ points in the plane) handles the upper part of that range. For the rest, one line $L_1$ is replaced by a projective plane of the least prime-power order $q$ with $p_k+1<q^2+q+1<p_k+p_k^{31/40+\varepsilon/2}$ (p. 134, (4)), from which points are deleted, keeping every line, until $p_k+1$ remain; its shortened lines are then broken up in turn (pp. 134--135).

Read depth

Claims checked: the setting, Theorem 1, the remark after it and Theorem 1* were read clause by clause on the page images of the print, and the proof of Theorem 1* was followed for structure. The extension from Theorem 1* to all $v$ is asserted, not written out, in the paper. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: the de Bruijn--Erdős theorem, Erdős's result on Grünbaum's problem, and the Heath-Brown--Iwaniec theorem on differences between consecutive primes.

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

None of the catalog's problems directly. The line counts just above $v$ that Problem 903 asks about lie outside the range of Theorem 1; Theorem 2 treats them.