1. FormalConjectures
  2. ErdosProblems
  3. «1034»
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Erdős Problem 1034

References:

open Filternamespace Erdos1034

JoinedToTwo G T Y holds when every vertex of Y is joined to at least two (distinct) vertices of T.

def JoinedToTwo {V : Type*} (G : SimpleGraph V) (T Y : Finset V) : Prop := ∀ y ∈ Y, ∃ u ∈ T, ∃ v ∈ T, u ≠ v ∧ G.Adj y u ∧ G.Adj y v

Let $G$ be a graph on $n$ vertices with $>n^2/4$ many edges. Must there be a triangle $T$ in $G$ and vertices $y_1,\ldots,y_t$, where $t>(\frac{1}{2}-o(1))n$, such that every $y_i$ is joined to at least two vertices of $T$?

A conjecture of Erdős and Faudree; a stronger version of [905].

This has been solved in the negative by Ma and Tang [MaTa25], who construct a graph with $n$ vertices and $>n^2/4$ edges in which every triangle has at most $(2-(5/2)^{1/2}+o(1))n$ vertices adjacent to at least two of its vertices (note that $2-(5/2)^{1/2}\approx 0.4189$).

@[category research solved, AMS 5, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos1034.lean"] theorem erdos_1034 : answer(False) ↔ ∀ ε : ℝ, 0 < ε → ∀ᶠ (n : ℕ) in atTop, ∀ G : SimpleGraph (Fin n), (n : ℝ) ^ 2 / 4 < (G.edgeSet.ncard : ℝ) → ∃ T : Finset (Fin n), G.IsNClique 3 T ∧ ∃ Y : Finset (Fin n), JoinedToTwo G T Y ∧ (1 / 2 - ε) * (n : ℝ) < (Y.card : ℝ) := ⊢ False ↔ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ∀ (G : SimpleGraph (Fin n)), ↑n ^ 2 / 4 < ↑G.edgeSet.ncard → ∃ T, G.IsNClique 3 T ∧ ∃ Y, JoinedToTwo G T Y ∧ (1 / 2 - ε) * ↑n < ↑Y.card All goals completed! 🐙

Erdős and Faudree asked about the threshold $h(n)$ such that every graph with $n$ vertices and $>n^2/4$ edges contained a triangle and $h(n)$ other vertices which are connected to at least two vertices of the triangle. The fact that every graph with $>n^2/4$ edges contains a book of size $n/6$ shows that $$(1/6-o(1))n \leq h(n).$$

@[category research solved, AMS 5] theorem erdos_1034.variants.lower_bound (ε : ℝ) (hε : 0 < ε) : ∀ᶠ (n : ℕ) in atTop, ∀ G : SimpleGraph (Fin n), (n : ℝ) ^ 2 / 4 < (G.edgeSet.ncard : ℝ) → ∃ T : Finset (Fin n), G.IsNClique 3 T ∧ ∃ Y : Finset (Fin n), JoinedToTwo G T Y ∧ (1 / 6 - ε) * (n : ℝ) ≤ (Y.card : ℝ) := ε:ℝhε:0 < ε⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (G : SimpleGraph (Fin n)), ↑n ^ 2 / 4 < ↑G.edgeSet.ncard → ∃ T, G.IsNClique 3 T ∧ ∃ Y, JoinedToTwo G T Y ∧ (1 / 6 - ε) * ↑n ≤ ↑Y.card All goals completed! 🐙

The construction of Ma and Tang [MaTa25] of a graph with $n$ vertices and $>n^2/4$ edges in which every triangle has at most $(2-(5/2)^{1/2}+o(1))n$ vertices adjacent to at least two of its vertices shows that, for the threshold $h(n)$ of erdos_1034.variants.lower_bound, $$h(n) \leq (2-(5/2)^{1/2}+o(1))n.$$

@[category research solved, AMS 5] theorem erdos_1034.variants.upper_bound (ε : ℝ) (hε : 0 < ε) : ∀ᶠ (n : ℕ) in atTop, ∃ G : SimpleGraph (Fin n), (n : ℝ) ^ 2 / 4 < (G.edgeSet.ncard : ℝ) ∧ ∀ T : Finset (Fin n), G.IsNClique 3 T → ∀ Y : Finset (Fin n), JoinedToTwo G T Y → (Y.card : ℝ) ≤ (2 - Real.sqrt (5 / 2) + ε) * (n : ℝ) := ε:ℝhε:0 < ε⊢ ∀ᶠ (n : ℕ) in atTop, ∃ G, ↑n ^ 2 / 4 < ↑G.edgeSet.ncard ∧ ∀ (T : Finset (Fin n)), G.IsNClique 3 T → ∀ (Y : Finset (Fin n)), JoinedToTwo G T Y → ↑Y.card ≤ (2 - √(5 / 2) + ε) * ↑n All goals completed! 🐙

Erdős suggested that the answer is different if $G$ has no $K_4$. In the comments Ma and Tang sketch a proof that the conjecture remains false even if we assume that $G$ contains no $K_4$, constructing a graph with $n$ vertices, $>n^2/4$ edges, and no $K_4$, in which every triangle has at most $(2\sqrt{3}-3+o(1))n$ vertices adjacent to at least two of its vertices (note that $2\sqrt{3}-3\approx 0.464$).

@[category research solved, AMS 5] theorem erdos_1034.variants.k4_free (ε : ℝ) (hε : 0 < ε) : ∀ᶠ (n : ℕ) in atTop, ∃ G : SimpleGraph (Fin n), G.CliqueFree 4 ∧ (n : ℝ) ^ 2 / 4 < (G.edgeSet.ncard : ℝ) ∧ ∀ T : Finset (Fin n), G.IsNClique 3 T → ∀ Y : Finset (Fin n), JoinedToTwo G T Y → (Y.card : ℝ) ≤ (2 * Real.sqrt 3 - 3 + ε) * (n : ℝ) := ε:ℝhε:0 < ε⊢ ∀ᶠ (n : ℕ) in atTop, ∃ G, G.CliqueFree 4 ∧ ↑n ^ 2 / 4 < ↑G.edgeSet.ncard ∧ ∀ (T : Finset (Fin n)), G.IsNClique 3 T → ∀ (Y : Finset (Fin n)), JoinedToTwo G T Y → ↑Y.card ≤ (2 * √3 - 3 + ε) * ↑n All goals completed! 🐙end Erdos1034