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Statement

A Steiner system with parameters $(n,q,r)$ is a set $S$ of $q$-subsets of an $n$-set $X$ such that every $r$-subset of $X$ lies in exactly one member of $S$; a design with parameters $(n,q,r,\lambda)$ is such a set in which every $r$-subset lies in exactly $\lambda$ members (p. 1). Counting the members of $S$ through a fixed $i$-subset of $X$ gives the necessary divisibility conditions: $\binom{q-i}{r-i}$ divides $\lambda\binom{n-i}{r-i}$ for every $0\le i\le r$ (p. 1). The Existence Conjecture, whose originator the paper says is not known, asserts that these conditions are also sufficient apart from finitely many exceptional $n$, for fixed $q$, $r$ and $\lambda$ (p. 1).

The paper proves the conjecture (abstract and p. 1); it has no numbered statement of it. On p. 2 it deduces the case $\lambda=1$ from Theorem 1.4 applied with $G=K_n^r$: for large $n$ the divisibility conditions are sufficient for the existence of Steiner systems. A Steiner system with parameters $(n,q,r)$ is the same as a $K_q^r$-decomposition of $K_n^r$ (p. 2), and for $G=K_n^r$ the $K_q^r$-divisibility of Definition 1.2 is the condition that $\binom{q-i}{r-i}$ divides $\binom{n-i}{r-i}$ for $0\le i\le r$. For general constant $\lambda$ the paper states that the existence of designs follows from Theorem 1.10 (p. 2), a design with parameters $(n,q,r,\lambda)$ being the same as a $K_q^r$-decomposition of the $r$-multigraph $\lambda\binom{[n]}r$.

The paper places the result in its history (pp. 1 and 4): the problem goes back to Plücker (1835), Kirkman (1846) and Steiner (1853); Wilson settled the case $r=2$; Hanani settled $(q,r)\in\{(4,2),(4,3),(5,2)\}$ for all $n$ and Kirkman the case $(3,2)$; before this paper only finitely many Steiner systems with $r\ge4$ were known, and it was not known whether any with $r\ge6$ exist.

Proof pointer

The deduction is the two remarks on p. 2 cited above, from Theorems 1.4 and 1.10; the paper does not write out the verification of their hypotheses for $K_n^r$ or for $\lambda\binom{[n]}r$.

Read depth

Claims checked: the definitions and the conjecture on p. 1, the remarks on p. 2 and the history on p. 4 were read clause by clause on the page images of the print. The proofs of Theorems 1.4 and 1.10 were not checked. Nothing here is independently reviewed.

Dependencies

Theorem 1.4 and Theorem 1.10.

Source. P. Keevash, The existence of designs, arXiv:1401.3665; the edition read and its page numbers are named on the source card.

Bears on

• Problem 722: with $k=q$, the problem asks whether for $k>r$ and $n$ large a Steiner system with parameters $(n,k,r)$ exists whenever $\binom{k-i}{r-i}$ divides $\binom{n-i}{r-i}$ for every $0\le i<r$. The case $\lambda=1$ of this result is that statement (the condition at $i=r$ holds trivially), so the paper answers the problem yes.