582.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 582

References:

erdosproblems.com/582
[Er75b] Erdős, Paul, Problems and results in combinatorial number theory. Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974) (1975), 295-310.
[Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968) (1969), 27-35.
[Er75d] Erdős, Paul, Problems and results on finite and infinite graphs. Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974) (1975), 183-192. (loose errata).
[FrRo86] Frankl, P. and Rödl, V., Large triangle-free subgraphs in graphs without {$K_4$}. Graphs Combin. (1986), 135-144.
[Sp88] Spencer, Joel, Three hundred million points suffice. J. Combin. Theory Ser. A (1988), 210-217.
[Lu07] Lu, Linyuan, Explicit construction of small Folkman graphs. SIAM J. Discrete Math. (2007), 1053-1060.
[DuRo08] Dudek, Andrzej and Rödl, Vojtěch, On the Folkman number {$f(2,3,4)$}. Experiment. Math. (2008), 63-67.
[RaXu07] Radziszowski, Stanisław P. and Xu, Xiaodong, On the most wanted Folkman graph. Geombinatorics (2007), 367-381.
[BiNe20] Bikov, Aleksandar and Nenov, Nedyalko, On the independence number of $(3,3)$-Ramsey graphs and the Folkman number $F_e(3,3;4)$. Australas. J. Combin. (2020), 35-50.
[ErHa67] Erdős, P. and Hajnal, A., Research Problem 2.5. J. Comb. Theory (1967).
[Fo70] Folkman, Jon, Graphs with monochromatic complete subgraphs in every edge coloring. SIAM J. Appl. Math. (1970), 19-24.
[LRX14] Lange, Alexander R. and Radziszowski, Stanisław P. and Xu, Xiaodong, Use of MAX-CUT for Ramsey arrowing of triangles. J. Combin. Math. Combin. Comput. (2014), 61-71.

Imports (1)

FormalConjecturesUtil

def Erdos582.EdgeRamseyTriangle A graph G is edge-Ramsey for triangles (in arrow notation line 60 source
Source documentation
A graph G is edge-Ramsey for triangles (in arrow notation, $G \to (K_3, K_3)^e$) if any $2$-colouring of the edges of G produces a monochromatic $K_3$: three pairwise adjacent vertices whose three connecting edges all receive the same colour.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph
Proof uses: Sym2, SimpleGraph.edgeSet, Fin, OfNat.ofNat, Nat, Exists, SimpleGraph.Adj, And, Eq, Subtype.mk, Membership.mem, Set, Sym2.mk
3 instances, coercions and auxiliary constants left out.
```
def EdgeRamseyTriangle {V : Type*} (G : SimpleGraph V) : Prop :=
…
```
theorem Erdos582.erdos_582 Does there exist a graph $G$ which contains no $K_4$, and yet any $2$-colouring of the edges produces a monochromatic $K_3$? sorry line 76 source
Source documentation
Does there exist a graph $G$ which contains no $K_4$, and yet any $2$-colouring of the edges produces a monochromatic $K_3$?
Erdős and Hajnal [ErHa67] first asked for the existence of any such graph. Existence was proved by Folkman [Fo70], but with very poor quantitative bounds. (As a result these quantities are often called Folkman numbers.) The current best bounds on the minimal number of vertices $N$ of such a graph are $21 \leq N \leq 786$, where the lower bound is due to Bikov and Nenov [BiNe20] and the upper bound is due to Lange, Radziszowski, and Xu [LRX14].

depends on sorry what this means
Statement uses: Iff, True, Exists, Fintype, SimpleGraph, And, SimpleGraph.CliqueFree, OfNat.ofNat, Nat, Erdos582.EdgeRamseyTriangle
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_582 : answer(True) ↔
    ∃ (V : Type*) (_ : Fintype V) (G : SimpleGraph V),
      G.CliqueFree 4 ∧ EdgeRamseyTriangle G := by
…
```
Mentions: Erdos582.EdgeRamseyTriangle