775.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 775

References:

erdosproblems.com/775
[Ga25] Gao, J., On cliques in hypergraphs. arXiv:2510.14804 (2025).
[MoMo65] Moon, J. W. and Moser, L., On cliques in graphs. Israel J. Math. (1965), 23-28.
[Sp71] Spencer, J. H., On cliques in graphs. Israel J. Math. (1971), 419-421.

Imports (1)

FormalConjecturesUtil

theorem Erdos775.erdos_775 Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different sizes of cliques (maximal complete subgraphs)? sorry line 45 source
Source documentation
Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different sizes of cliques (maximal complete subgraphs)?
The answer is no, as proved by Gao [Ga25]: more generally, for any $k\geq 3$, every $k$-uniform hypergraph on $n$ vertices contains at most $n-f_k(n)$ different sizes of cliques, where $f_k(n)\to \infty$ as $n\to \infty$.

depends on sorry what this means
Statement uses: Iff, False, Exists, Nat, Filter.Frequently, UniformHypergraph, Fin, OfNat.ofNat, LE.le, HSub.hSub, Set.ncard, UniformHypergraph.cliqueSizes, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_775 : answer(False) ↔
    ∃ C : ℕ, ∃ᶠ n : ℕ in atTop, ∃ H : UniformHypergraph (Fin n) 3,
      n - C ≤ (UniformHypergraph.cliqueSizes H).ncard := by
…
```
Mentions: UniformHypergraph, UniformHypergraph.cliqueSizes
theorem Erdos775.erdos_775.variants.gao Documented declaration: gao sorry line 56 source
Source documentation
The answer is no, as proved by Gao [Ga25]: more generally, for any $k\geq 3$, every $k$-uniform hypergraph on $n$ vertices contains at most $n-f_k(n)$ different sizes of cliques, where $f_k(n)\to \infty$ as $n\to \infty$.

depends on sorry what this means
Statement uses: Exists, Nat, And, Filter.Tendsto, Filter.atTop, Filter.Eventually, UniformHypergraph, Fin, OfNat.ofNat, LE.le, HAdd.hAdd, Set.ncard, UniformHypergraph.cliqueSizes
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_775.variants.gao :
    ∃ f : ℕ → ℕ, Tendsto f atTop atTop ∧
      ∀ᶠ n : ℕ in atTop, ∀ H : UniformHypergraph (Fin n) 3,
        (UniformHypergraph.cliqueSizes H).ncard + f n ≤ n := by
…
```
Mentions: UniformHypergraph, UniformHypergraph.cliqueSizes
theorem Erdos775.erdos_775.variants.spencer For graphs, Spencer [Sp71] constructed a graph which contains cliques of at least $n-\log_2n+O(1)$ different sizes sorry line 67 source
Source documentation
For graphs, Spencer [Sp71] constructed a graph which contains cliques of at least $n-\log_2n+O(1)$ different sizes.

depends on sorry what this means
Statement uses: Exists, Real, Nat, SimpleGraph, Fin, LE.le, HSub.hSub, Real.logb, OfNat.ofNat, Set.ncard, SimpleGraph.cliqueSizes
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_775.variants.spencer :
    ∃ C : ℝ, ∀ n : ℕ, ∃ G : SimpleGraph (Fin n),
      (n : ℝ) - Real.logb 2 (n : ℝ) - C ≤ ((SimpleGraph.cliqueSizes G).ncard : ℝ) := by
…
```
Mentions: SimpleGraph.cliqueSizes
theorem Erdos775.erdos_775.variants.moon_moser For graphs, Spencer [Sp71] constructed a graph which contains cliques of at least $n-\log_2n+O(1)$ different sizes, which Moon and Moser [MoMo65] showed to be best possible sorry line 77 source
Source documentation
For graphs, Spencer [Sp71] constructed a graph which contains cliques of at least $n-\log_2n+O(1)$ different sizes, which Moon and Moser [MoMo65] showed to be best possible.

depends on sorry what this means
Statement uses: Exists, Real, Nat, SimpleGraph, Fin, LE.le, Set.ncard, SimpleGraph.cliqueSizes, HAdd.hAdd, HSub.hSub, Real.logb, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_775.variants.moon_moser :
    ∃ C : ℝ, ∀ n : ℕ, ∀ G : SimpleGraph (Fin n),
      ((SimpleGraph.cliqueSizes G).ncard : ℝ) ≤ (n : ℝ) - Real.logb 2 (n : ℝ) + C := by
…
```
Mentions: SimpleGraph.cliqueSizes