904.lean`

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

3 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 904

References:

erdosproblems.com/904
[BoEr75] Bollobás, B. and Erdős, P., Unsolved Problems. Proc. Fifth British Combinatorial Conf. (1975), 678-680.
[Er75] Erdős, P., Some recent progress on extremal problems in graph theory. Congr. Numer. (1975), 3-14.
[Ed78] Edwards, C. S., Complete subgraphs with largest sum of vertex degrees. (1978), 293-306.
[Fa92] Faudree, Ralph J., Complete subgraphs with large degree sums. J. Graph Theory (1992), 327-334.
[BoNi05] Bollobás, Béla and Nikiforov, Vladimir, The sum of degrees in cliques. Electron. J. Combin. (2005), Note 21, 10.

Imports (1)

FormalConjecturesUtil

abbrev Erdos904.n An abbreviation for the fixed number of vertices $n$ in the graph line 39 source
Source documentation
An abbreviation for the fixed number of vertices $n$ in the graph.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Fintype, Nat
Proof uses: Fintype.card
```
abbrev n (V : Type*) [Fintype V] : ℕ := Fintype.card V
```
abbrev Erdos904.turanNumber The number of edges of the Turán graph $T(n, r)$, i.e. the Turán number line 42 source
Source documentation
The number of edges of the Turán graph $T(n, r)$, i.e. the Turán number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.card, Sym2, Fin, SimpleGraph.edgeFinset, SimpleGraph.turanGraph
4 instances, coercions and auxiliary constants left out.
```
abbrev turanNumber (n r : ℕ) : ℕ := #(SimpleGraph.turanGraph n r).edgeFinset
…
```
Mentions: Erdos904.n
theorem Erdos904.erdos_904 Let $r\geq 2$ and let $t_r(n)$ be the Turán number (the maximal number of edges in a graph on $n$ vertices with no $K_{r+1}$) sorry line 54 source
Source documentation
Let $r\geq 2$ and let $t_r(n)$ be the Turán number (the maximal number of edges in a graph on $n$ vertices with no $K_{r+1}$).
If $G$ is a graph with $n$ vertices and $m\geq t_r(n)$ edges there exists a clique on $r$ vertices, say $x_1,\ldots,x_r$, such that $$d(x_1)+\cdots+d(x_r)\geq \frac{2rm}{n}.$$
A conjecture of Bollobás and Erdős. This was conjectured in [Er75] only in the special case $r=3$. Edwards [Ed78] proved the conjecture for $2\leq r\leq 8$ (under the additional assumption that $n\geq r^2$). Faudree [Fa92] proved the conjecture for all $r\geq 2$ provided $n>\frac{r-1}{4}r^2$. The full conjecture was proved by Bollobás and Nikiforov [BoNi05].

depends on sorry what this means
Statement uses: Iff, True, Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, Nat, Membership.mem, Set, Set.Icc, OfNat.ofNat, Erdos904.n, LE.le, Erdos904.turanNumber, Finset.card, Sym2, SimpleGraph.edgeFinset, Exists, Finset, And and 5 more
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_904 :
    answer(True) ↔ ∀ (V : Type*) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj]
      (r : ℕ) (hr : r ∈ Set.Icc 1 (n V)) (hm : turanNumber (n V) r ≤ #G.edgeFinset),
      ∃ s, G.IsNClique r s ∧ 2 * r * #G.edgeFinset ≤ n V * ∑ v ∈ s, G.degree v := by
…
```
Mentions: Erdos904.n, Erdos904.turanNumber