545.lean`

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

2 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 545

References:

erdosproblems.com/545
[ErGr75] Erdős, P. and Graham, R. L., On partition theorems for finite graphs. Infinite and finite sets (1975), 515-527.
[Er84b] Erdős, P., On some problems in graph theory, combinatorial analysis and combinatorial number theory. Graph theory and combinatorics (Cambridge, 1983) (1984), 1-17.

Imports (1)

FormalConjecturesUtil

def Erdos545.knPlusTEdges The graph formed by connecting a new vertex (none) to $t$ of the vertices of $K_n$ (Fin n) line 40 source
Source documentation
The graph formed by connecting a new vertex (none) to $t$ of the vertices of $K_n$ (Fin n).

standard axioms: propext what this means
Statement uses: Nat, SimpleGraph, Option, Fin
Proof uses: SimpleGraph.mk, Erdos545.knPlusTEdges.match_1, Ne, LT.lt, Unit, False
4 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
def knPlusTEdges (n t : ℕ) : SimpleGraph (Option (Fin n)) where
…
```
theorem Erdos545.erdos_545 Let $m$ be sufficiently large and let $G$ be a graph with $m$ edges and no isolated vertices sorry line 65 source
Source documentation
Let $m$ be sufficiently large and let $G$ be a graph with $m$ edges and no isolated vertices. Is the Ramsey number $R(G)$ maximised when $G$ is 'as complete as possible'? That is, if $m=\binom{n}{2}+t$ edges with $0\leq t < n$ then is $$R(G)\leq R(H),$$ where $H$ is the graph formed by connecting a new vertex to $t$ of the vertices of $K_n$?
A question of Erdős and Graham. The restriction to sufficiently large $m$ excludes the small counterexamples recorded on the source page.
This problem is #10 in Ramsey Theory in the graphs problem collection.

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, LT.lt, Eq, HAdd.hAdd, Nat.choose, OfNat.ofNat, Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, Set.ncard, Sym2, SimpleGraph.edgeSet and 6 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_545 : answer(sorry) ↔
    ∀ᶠ m : ℕ in atTop, ∀ (n t : ℕ), t < n → m = n.choose 2 + t →
      ∀ (V : Type) [Fintype V] (G : SimpleGraph V) [DecidableRel G.Adj],
        (∀ v, 0 < G.degree v) →
        G.edgeSet.ncard = m →
        SimpleGraph.diagonalGraphRamsey G ≤
          SimpleGraph.diagonalGraphRamsey (knPlusTEdges n t) := by
…
```
Mentions: SimpleGraph.diagonalGraphRamsey, Erdos545.knPlusTEdges