Erdős Problem 1020

formal-conjectures · erdos-problems · ams-5

Let $f(n;r,k)$ be the maximal number of edges in an $r$-uniform hypergraph which contains no set of $k$ many independent edges.

For all $r\geq 3$, $$f(n;r,k)=\max\left(\binom{rk-1}{r}, \binom{n}{r}-\binom{n-k+1}{r}\right).$$

Note: the source states the formula with no range on n or k, but some restriction
is needed: e.g. for r = 3, k = 2, n = 4 no two disjoint triples fit in 4
vertices, so the left-hand side is 4.choose 3 = 4 while the right-hand side is
5.choose 3 = 10. We require k ≥ 1 and n ≥ r*k - 1: this is the smallest n
accommodating the construction counted by the first term (all r-subsets of a fixed
(r*k - 1)-set), and at n = r*k - 1 the equality holds trivially, since the complete
r-uniform hypergraph has no k-matching. The source's commentary likewise calls the
case n < k*r trivial.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://www.erdosproblems.com/1020
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/ErdosProblems/1020.lean
Local target: Corpus.Erdos1020.erdos_1020
Source SHA-256: 001c70d5082e1bf94d6c3dcf1beb02998e14452d8f40ced9b3af081bc52c3236
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record