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The high-girth existence conjecture for Steiner systems

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A Steiner system with parameters (n,q,r)(n,q,r) is a design in which every rr-subset belongs to exactly one qq-subset. For a partial (n,q,r)(n,q,r)-Steiner system, the girth is the smallest integer g≥2g\geq2 for which it has a ((q−r)g+r,g)((q-r)g+r,g)-configuration. High-girth existence conjecture. For all integers q>r≥2q > r \geq 2 and every integer g≥2g\ge 2, there exists n0n_0 such that for all n≥n0n\ge n_0 satisfying

(q−ir−i) ∣ (n−ir−i)\binom{q-i}{r-i}~|~\binom{n-i}{r-i}

for all 0≤i≤r−10\le i \le r-1, there exists an (n,q,r)(n,q,r)-Steiner system with girth at least gg. This is presented as a common generalization of the design Existence Conjecture and Erdős' high-girth Steiner triple-system conjecture. The paper's title and abstract state that it proves this conjecture via refined absorption.

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  1. High-girth existence conjecture for Steiner systems

    Let q>r≥2q>r\geq 2 and let a girth-gg decomposition mean a KqrK_q^r-decomposition containing no configuration of the forbidden size below gg. High Girth Existence Conjecture. For every integer g≥3g\geq 3, every sufficiently large KqrK_q^r-divisible complete rr-uniform hypergraph admits a KqrK_q^r-decomposition with girth at least gg. This generalizes the high-girth Steiner triple system conjecture; the source gives no resolution status for the general hypergraph statement.

    source: Luke Postle, “Refined Absorption: A New Proof of the Existence Conjecture and its Applications to Extremal and Probabilistic Design Theory”, arXiv:2510.19978 (2025).

References

Primary source

Michelle Delcourt and Luke Postle, “Proof of the High Girth Existence Conjecture via Refined Absorption”, arXiv:2402.17856 (2024).

design-theoryextremal-combinatoricshigh-girth-combinatoricshypergraph-decompositionssteiner-systems
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