Erdős Problem 595

References:

• erdosproblems.com/595
• [Er87] Erdős, Paul, Problems and results on set systems and hypergraphs. Extremal problems for finite sets (Visegrád, 1991), Bolyai Soc. Math. Stud. (1994), 217-227.
• [Fo70] Folkman, Jon, Graphs with monochromatic complete subgraphs in every edge coloring. SIAM J. Appl. Math. (1970), 19:340-345.
• [NeRo75] Nešetřil, Jaroslav and Rödl, Vojtěch, Type theory of partition problems of graphs. Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), Academia, Prague (1975), 405-412.

def Erdos595.IsCountableUnionOfTriangleFree {V : Type u_1} (G : SimpleGraph V) : Prop

Equations

• Erdos595.IsCountableUnionOfTriangleFree G = ∃ (H : ℕ → SimpleGraph V), (∀ (i : ℕ), (H i).CliqueFree 3) ∧ G = ⨆ (i : ℕ), H i

theorem Erdos595.erdos_595 : True ↔ ∃ (V : Type u_1) (_ : Infinite V) (G : SimpleGraph V), G.CliqueFree 4 ∧ ¬IsCountableUnionOfTriangleFree G

Erdős Problem 595 ($250): Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs?

A problem of Erdős and Hajnal [Er87].

theorem Erdos595.erdos_595.variants.folkman_finite : True ↔ ∀ (n : ℕ), 1 ≤ n → ∃ (V : Type u_1) (x : Fintype V) (G : SimpleGraph V), G.CliqueFree 4 ∧ ∀ (H : Fin n → SimpleGraph V), (∀ (i : Fin n), (H i).CliqueFree 3) → G ≠ ⨆ (i : Fin n), H i

Folkman–Nešetřil–Rödl (finite version) [Fo70, NeRo75]: For every n ≥ 1, there exists a graph G (on a finite vertex set) that contains no $K_4$ and whose edges cannot be covered by n triangle-free graphs.

More precisely: for every n : ℕ with 1 ≤ n, there exist a finite type V and a graph G : SimpleGraph V with:

1. G.CliqueFree 4 (no $K_4$), and
2. For every family H : Fin n → SimpleGraph V of triangle-free graphs, G ≠ ⨆ i, H i.

This is the finite analogue of Problem 595. The proofs of Folkman [Fo70] and Nešetřil–Rödl [NeRo75] give different explicit constructions.

theorem Erdos595.erdos_595.variants.subgraph_of_countable_union {V : Type u_1} {G H : SimpleGraph V} (hH : H ≤ G) (hG : IsCountableUnionOfTriangleFree G) : IsCountableUnionOfTriangleFree H

Monotonicity: If G is a countable union of triangle-free graphs and H ≤ G (i.e., H is a subgraph of G), then H is also a countable union of triangle-free graphs.

Proof: If G = ⨆ i, G_i with each G_i triangle-free, then H = ⨆ i, H ⊓ G_i. Each H ⊓ G_i is triangle-free because it is a subgraph of G_i.

theorem Erdos595.erdos_595.variants.triangle_free_is_union {V : Type u_1} (G : SimpleGraph V) (hG : G.CliqueFree 3) : IsCountableUnionOfTriangleFree G

Triangle-free graphs are trivially countable unions of triangle-free graphs: if G is already triangle-free, then G = ⨆ i : ℕ, G_i where G_0 = G and G_i = ⊥ for i ≥ 1.

theorem Erdos595.erdos_595.variants.complete_nat_is_union : IsCountableUnionOfTriangleFree ⊤

The complete graph ⊤ on ℕ is a countable union of triangle-free graphs: we decompose it into the family of star graphs {H_m}_{m : ℕ}, where H_m is the graph with edges {m, n} for all n ≠ m. Each star is triangle-free (any two non-center vertices share no edge within the star), and their union covers all edges of ⊤.

Proof sketch (star triangle-free): If {a, b, c} were a triangle in H_m, then each of the three edges {a, b}, {a, c}, {b, c} would pass through m. In particular, from {a, b} we get a = m or b = m; from {b, c} we get b = m or c = m. Case analysis shows that two vertices must equal m, contradicting the triangle having three distinct vertices.

theorem Erdos595.erdos_595.variants.K4_not_cliqueFree : ¬⊤.CliqueFree 4

The complete graph ⊤ on Fin 4 is not $K_4$-free: ⊤ on Fin 4 equals the complete graph $K_4$, so it contains $K_4$ as a subgraph and is not $K_4$-free.

This sanity check confirms the $K_4$-free hypothesis of Problem 595 is non-trivial.

theorem Erdos595.erdos_595.variants.reformulation_edge_colouring {V : Type u_1} (G : SimpleGraph V) : IsCountableUnionOfTriangleFree G ↔ ∃ (c : ↑G.edgeSet → ℕ), ∀ (n : ℕ), (SimpleGraph.fromEdgeSet {e : Sym2 V | ∃ (h : e ∈ G.edgeSet), c ⟨e, h⟩ = n}).CliqueFree 3

Reformulation via edge colourings: A graph G is a countable union of triangle-free graphs if and only if there is a colouring of the edges of G by ℕ such that no monochromatic triangle exists.

More precisely: IsCountableUnionOfTriangleFree G is equivalent to the existence of a map c : G.edgeSet → ℕ such that for each n : ℕ, the subgraph of edges coloured n is triangle-free.