1034.lean`

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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 1034

References:

erdosproblems.com/1034
[Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350.
[MaTa25] Ma, Jie and Tang, Quanyu, On Erdős problem #1034. staff.ustc.edu.cn/~jiema/Erdos-1034.pdf

Imports (1)

FormalConjecturesUtil

def Erdos1034.JoinedToTwo JoinedToTwo G T Y holds when every vertex of Y is joined to at least two (distinct) vertices of T line 41 source
Source documentation
JoinedToTwo G T Y holds when every vertex of Y is joined to at least two (distinct) vertices of T.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Finset
Proof uses: Membership.mem, Exists, And, Ne, SimpleGraph.Adj
2 instances, coercions and auxiliary constants left out.
```
def JoinedToTwo {V : Type*} (G : SimpleGraph V) (T Y : Finset V) : Prop :=
…
```
theorem Erdos1034.erdos_1034 Let $G$ be a graph on $n$ vertices with $>n^2/4$ many edges sorry line 56 source
Source documentation
Let $G$ be a graph on $n$ vertices with $>n^2/4$ many edges. Must there be a triangle $T$ in $G$ and vertices $y_1,\ldots,y_t$, where $t>(\frac{1}{2}-o(1))n$, such that every $y_i$ is joined to at least two vertices of $T$?
A conjecture of Erdős and Faudree; a stronger version of [905].
This has been solved in the negative by Ma and Tang [MaTa25], who construct a graph with $n$ vertices and $>n^2/4$ edges in which every triangle has at most $(2-(5/2)^{1/2}+o(1))n$ vertices adjacent to at least two of its vertices (note that $2-(5/2)^{1/2}\approx 0.4189$).

depends on sorry what this means
Statement uses: Iff, False, Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, SimpleGraph, Fin, HDiv.hDiv, HPow.hPow, Set.ncard, Sym2, SimpleGraph.edgeSet, Exists, Finset, And, SimpleGraph.IsNClique, Erdos1034.JoinedToTwo, HMul.hMul and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1034 : answer(False) ↔
    ∀ ε : ℝ, 0 < ε → ∀ᶠ (n : ℕ) in atTop, ∀ G : SimpleGraph (Fin n),
      (n : ℝ) ^ 2 / 4 < (G.edgeSet.ncard : ℝ) →
        ∃ T : Finset (Fin n), G.IsNClique 3 T ∧ ∃ Y : Finset (Fin n),
          JoinedToTwo G T Y ∧ (1 / 2 - ε) * (n : ℝ) < (Y.card : ℝ) := by
…
```
Mentions: Erdos1034.JoinedToTwo
theorem Erdos1034.erdos_1034.variants.lower_bound Erdős and Faudree asked about the threshold sorry line 71 source
Source documentation
Erdős and Faudree asked about the threshold $h(n)$ such that every graph with $n$ vertices and $>n^2/4$ edges contained a triangle and $h(n)$ other vertices which are connected to at least two vertices of the triangle. The fact that every graph with $>n^2/4$ edges contains a book of size $n/6$ shows that $$(1/6-o(1))n \leq h(n).$$

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, SimpleGraph, Fin, HDiv.hDiv, HPow.hPow, Set.ncard, Sym2, SimpleGraph.edgeSet, Exists, Finset, And, SimpleGraph.IsNClique, Erdos1034.JoinedToTwo, LE.le, HMul.hMul, HSub.hSub and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1034.variants.lower_bound (ε : ℝ) (hε : 0 < ε) :
    ∀ᶠ (n : ℕ) in atTop, ∀ G : SimpleGraph (Fin n),
      (n : ℝ) ^ 2 / 4 < (G.edgeSet.ncard : ℝ) →
        ∃ T : Finset (Fin n), G.IsNClique 3 T ∧ ∃ Y : Finset (Fin n),
          JoinedToTwo G T Y ∧ (1 / 6 - ε) * (n : ℝ) ≤ (Y.card : ℝ) := by
…
```
Mentions: Erdos1034.JoinedToTwo
theorem Erdos1034.erdos_1034.variants.upper_bound Documented declaration: upper_bound sorry line 85 source
Source documentation
The construction of Ma and Tang [MaTa25] of a graph with $n$ vertices and $>n^2/4$ edges in which every triangle has at most $(2-(5/2)^{1/2}+o(1))n$ vertices adjacent to at least two of its vertices shows that, for the threshold $h(n)$ of erdos_1034.variants.lower_bound, $$h(n) \leq (2-(5/2)^{1/2}+o(1))n.$$

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Exists, SimpleGraph, Fin, And, HDiv.hDiv, HPow.hPow, Set.ncard, Sym2, SimpleGraph.edgeSet, Finset, SimpleGraph.IsNClique, Erdos1034.JoinedToTwo, LE.le, Finset.card, HMul.hMul and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1034.variants.upper_bound (ε : ℝ) (hε : 0 < ε) :
    ∀ᶠ (n : ℕ) in atTop, ∃ G : SimpleGraph (Fin n),
      (n : ℝ) ^ 2 / 4 < (G.edgeSet.ncard : ℝ) ∧
        ∀ T : Finset (Fin n), G.IsNClique 3 T → ∀ Y : Finset (Fin n),
          JoinedToTwo G T Y → (Y.card : ℝ) ≤ (2 - Real.sqrt (5 / 2) + ε) * (n : ℝ) := by
…
```
Mentions: Erdos1034.JoinedToTwo
theorem Erdos1034.erdos_1034.variants.k4_free Erdős suggested that the answer is different if $G$ has no $K_4$ sorry line 100 source
Source documentation
Erdős suggested that the answer is different if $G$ has no $K_4$. In the comments Ma and Tang sketch a proof that the conjecture remains false even if we assume that $G$ contains no $K_4$, constructing a graph with $n$ vertices, $>n^2/4$ edges, and no $K_4$, in which every triangle has at most $(2\sqrt{3}-3+o(1))n$ vertices adjacent to at least two of its vertices (note that $2\sqrt{3}-3\approx 0.464$).

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Exists, SimpleGraph, Fin, And, SimpleGraph.CliqueFree, HDiv.hDiv, HPow.hPow, Set.ncard, Sym2, SimpleGraph.edgeSet, Finset, SimpleGraph.IsNClique, Erdos1034.JoinedToTwo, LE.le, Finset.card and 5 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1034.variants.k4_free (ε : ℝ) (hε : 0 < ε) :
    ∀ᶠ (n : ℕ) in atTop, ∃ G : SimpleGraph (Fin n), G.CliqueFree 4 ∧
      (n : ℝ) ^ 2 / 4 < (G.edgeSet.ncard : ℝ) ∧
        ∀ T : Finset (Fin n), G.IsNClique 3 T → ∀ Y : Finset (Fin n),
          JoinedToTwo G T Y → (Y.card : ℝ) ≤ (2 * Real.sqrt 3 - 3 + ε) * (n : ℝ) := by
…
```
Mentions: Erdos1034.JoinedToTwo