667.lean`

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7 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 667

A problem of Erdős, Faudree, Rousseau, and Schelp on the growth exponent of the largest clique that is guaranteed inside a locally dense graph.

References:

erdosproblems.com/667
[Er97f] P. Erdős, Some unsolved problems. In: Combinatorics, Geometry and Probability (Cambridge, 1993), Cambridge Univ. Press (1997), 1-10.
[EFRS94] P. Erdős, R. J. Faudree, C. C. Rousseau and R. H. Schelp, A local density condition for triangles. Discrete Math. 127 (1994), 153-161.

Imports (1)

FormalConjecturesUtil

def Erdos667.LocallyDense Documented declaration: LocallyDense line 43 source
Source documentation
A graph G on n vertices is (p, q)-locally dense if every set of p of its vertices spans at least q edges of G; equivalently, every induced subgraph on p vertices has at least q edges.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, SimpleGraph, Fin
Proof uses: Finset, Eq, Finset.card, LE.le, Set.ncard, Sym2, SimpleGraph.edgeSet, SimpleGraph.induce
4 instances, coercions and auxiliary constants left out.
```
def LocallyDense (p q : ℕ) {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
…
```
def Erdos667.H H p q n is H(n) = H(n; p, q): the largest m such that every (p, q)-locally dense graph on n vertices must contain a complete graph on m vertices Kₘ line 52 source
Source documentation
H p q n is H(n) = H(n; p, q): the largest m such that every (p, q)-locally dense graph on n vertices must contain a complete graph on m vertices Kₘ.
The condition "G contains Kₘ" is ¬ G.CliqueFree m. Any m-clique needs m ≤ n vertices, and the property is downward closed in m, so Nat.findGreatest _ n returns the genuine maximum.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Nat.findGreatest, SimpleGraph, Fin, Erdos667.LocallyDense, Not, SimpleGraph.CliqueFree, Classical.propDecidable
6 instances, coercions and auxiliary constants left out.
```
noncomputable def H (p q n : ℕ) : ℕ :=
…
```
def Erdos667.c The growth exponent c(p, q) = liminf_{n → ∞} (log H(n)) / (log n) line 57 source
Source documentation
The growth exponent c(p, q) = liminf_{n → ∞} (log H(n)) / (log n).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Filter.liminf, HDiv.hDiv, Real.log, Erdos667.H, Filter.atTop
8 instances, coercions and auxiliary constants left out.
```
noncomputable def c (p q : ℕ) : ℝ :=
…
```
theorem Erdos667.erdos_667 Let $p, q \geq 1$ be fixed integers sorry line 78 source
Source documentation
Let $p, q \geq 1$ be fixed integers. Define $H(n) = H(n; p, q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$ vertices spans at least $q$ edges must contain a complete graph on $m$ vertices. Is $$ c(p, q) = \liminf_{n \to \infty} \frac{\log H(n)}{\log n} $$ a strictly increasing function of $q$ for $1 \leq q \leq \binom{p-1}{2} + 1$?
A problem of Erdős, Faudree, Rousseau, and Schelp. When $q = 1$ this corresponds exactly to the classical Ramsey problem, and hence $\tfrac{1}{p-1} \leq c(p, 1) \leq \tfrac{2}{p+1}$ for $p \geq 3$. It is easy to see that if $q = \binom{p-1}{2} + 1$ then $c(p, q) = 1$.
erdosproblems.com/667 also states that Erdős, Faudree, Rousseau and Schelp showed $c(p, \binom{p-1}{2}) \leq 1/2$. This cannot hold for all $p$: for $p = 6$, $q = 10$, the complement of a $(6, 10)$-locally dense graph is $C_6$-free, so Bondy–Simonovits together with $\alpha(F) \geq n^2 / (2e(F) + n)$ gives $c(6, 10) \geq 2/3$. That remark is therefore not formalised here.

depends on sorry what this means
Statement uses: Iff, True, Nat, StrictMonoOn, Real, Erdos667.c, Set.Icc, OfNat.ofNat, HAdd.hAdd, Nat.choose, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_667 :
    answer(sorry) ↔
      ∀ p : ℕ, StrictMonoOn (c p) (Set.Icc 1 (Nat.choose (p - 1) 2 + 1)) := by
…
```
Mentions: Erdos667.c
theorem Erdos667.erdos_667.variants.monotoneOn Documented declaration: monotoneOn sorry line 89 source
Source documentation
That $c(p, q)$ is nondecreasing in $q$ for fixed $p$ is a straightforward consequence of the definitions: enlarging $q$ shrinks the family of $(p, q)$-locally dense graphs, which can only increase the guaranteed clique size $H(n)$, and hence the exponent.

depends on sorry what this means
Statement uses: Nat, MonotoneOn, Real, Erdos667.c, Set.Icc, OfNat.ofNat, HAdd.hAdd, Nat.choose, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_667.variants.monotoneOn (p : ℕ) :
    MonotoneOn (c p) (Set.Icc 1 (Nat.choose (p - 1) 2 + 1)) := by
…
```
Mentions: Erdos667.c
theorem Erdos667.erdos_667.variants.endpoint The right endpoint: when $q = \binom{p-1}{2} + 1$ one has $c(p, q) = 1$ (stated as "easy to see" on erdosproblems.com/667) sorry line 98 source
Source documentation
The right endpoint: when $q = \binom{p-1}{2} + 1$ one has $c(p, q) = 1$ (stated as "easy to see" on erdosproblems.com/667).

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Eq, Real, Erdos667.c, HAdd.hAdd, Nat.choose, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_667.variants.endpoint (p : ℕ) (hp : 2 ≤ p) :
    c p (Nat.choose (p - 1) 2 + 1) = 1 := by
…
```
Mentions: Erdos667.c
theorem Erdos667.erdos_667.variants.ramsey_bounds The case $q = 1$ is the classical Ramsey problem, giving the bounds $\tfrac{1}{p-1} \leq c(p, 1) \leq \tfrac{2}{p+1}$ for $p \geq 3$ sorry line 108 source
Source documentation
The case $q = 1$ is the classical Ramsey problem, giving the bounds $\tfrac{1}{p-1} \leq c(p, 1) \leq \tfrac{2}{p+1}$ for $p \geq 3$. (For $p = 2$ the condition forces $G$ to be complete, so $c(2, 1) = 1$ and the upper bound fails.)

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, And, Real, HDiv.hDiv, HSub.hSub, Erdos667.c, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_667.variants.ramsey_bounds (p : ℕ) (hp : 3 ≤ p) :
    1 / ((p : ℝ) - 1) ≤ c p 1 ∧ c p 1 ≤ 2 / ((p : ℝ) + 1) := by
…
```
Mentions: Erdos667.c