Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Combinatorics P-erdos-1020

Erdős Problem #1020

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≥ 3, f(n;r,k)=max(C(rk-1, r), C(n, r)-C(n-k+1, r)). Note: the source states the formula with no range on n or k, but some restriction is needed: e.g.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

Start working on it Submit a claim Follow
Cite 
@misc{cairn-erdos-1020,
  title        = {Erdős Problem #1020},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1020}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Let be the maximal number of edges in an -uniform hypergraph which contains no set of many independent edges.

For all ,

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.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1020».

theorem erdos_1020 (r : ℕ) (hr : 3 ≤ r) (n k : ℕ) (hk : 0 < k)
    (hrk : r * k - 1 ≤ n) :
    f n r k = max ((r * k - 1).choose r)
      (n.choose r - (n - k + 1).choose r)

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1020. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

  • erdosproblems.com/1020
  • [BDE76] Bollobás, B. and Daykin, D. E. and Erdős, P., Sets of independent edges of a hypergraph. Quart. J. Math. Oxford Ser. (2) (1976), 25--32.
  • [Er65d] Erdős, P., A problem on independent {}-tuples. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (1965), 93--95.
  • [ErGa59] Erdős, P. and Gallai, T., On maximal paths and circuits of graphs. Acta Math. Acad. Sci. Hungar. (1959), 337-356 (unbound insert).
  • [FLM12] Frankl, Peter and Łuczak, Tomasz and Mieczkowska, Katarzyna, *On matchings in hypergraphs*. Electron. J. Combin. (2012), Paper 42, 5.
  • [FRR12] Frankl, Peter and Rödl, Vojtech and Ruciński, Andrzej, *On the maximum number of edges in a triple system not containing a disjoint family of a given size*. Combin. Probab. Comput. (2012), 141--148.
  • [Fr17] Frankl, Peter, Proof of the {E}rdős matching conjecture in a new range. Israel J. Math. (2017), 421--430.
  • [Fr87] Frankl, Peter, The shifting technique in extremal set theory. (1987), 81--110.
  • [HLS12] Huang, Hao and Loh, Po-Shen and Sudakov, Benny, *The size of a hypergraph and its matching number*. Combin. Probab. Comput. (2012), 442--450.
  • [Kl68] Kleitman, Daniel J., *Maximal number of subsets of a finite set no {} of which are pairwise disjoint*. J. Combinatorial Theory (1968), 157--163.
  • [KoKu23] Kolupaev, Dmitriy and Kupavskii, Andrey, *Erdős matching conjecture for almost perfect matchings*. Discrete Math. (2023), Paper No. 113304, 9.
  • [LuMi14] Łuczak, Tomasz and Mieczkowska, Katarzyna, *On {E}rdős' extremal problem on matchings in hypergraphs*. J. Combin. Theory Ser. A (2014), 178--194.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.