639.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 639

References:

erdosproblems.com/639
[Er97d] Erdős, Paul, Some recent problems and results in graph theory. Discrete Math. (1997), 81-85.
[KeSu04] Keevash, Peter and Sudakov, Benny, On the number of edges not covered by monochromatic copies of a fixed graph. J. Combin. Theory Ser. B (2004), 41-53.
[Py86] Pyber, L., Clique covering of graphs. Combinatorica (1986), 393-398.

Imports (1)

FormalConjecturesUtil

theorem Erdos639.erdos_639 Is it true that if the edges of $K_n$ are 2-coloured then there are at most $n^2/4$ many edges which do not occur in a monochromatic triangle? sorry line 62 source
Source documentation
Is it true that if the edges of $K_n$ are 2-coloured then there are at most $n^2/4$ many edges which do not occur in a monochromatic triangle?
Solved by Erdős, Rousseau, and Schelp for large $n$, but unpublished. Alon has observed that this also follows from a result of Pyber [Py86], which states that (for large enough $n$) at most $\lfloor n^2/4\rfloor+2$ monochromatic cliques cover all edges of a $2$-coloured $K_n$.
This problem was solved completely by Keevash and Sudakov [KeSu04], who proved that the correct threshold is $\lfloor n^2/4\rfloor$ for all $n\geq 7$, is $\binom{n}{2}$ for $n\leq 5$, and is $10$ for $n=6$.
Since the bound fails for small $n$ (at $n=6$ the threshold is $10 > 6^2/4$), the statement is formalized in the asymptotic reading in which the problem was posed and solved: for all sufficiently large $n$, every $2$-colouring of the edges of $K_n$ leaves at most $n^2/4$ edges not occurring in a monochromatic triangle. Edges of $K_n$ are the non-diagonal unordered pairs Sym2 (Fin n); an edge $\{x, y\}$ occurs in a monochromatic triangle if and only if there is a third vertex $z$ with $C(\{x, z\}) = C(\{y, z\}) = C(\{x, y\})$.
The linked file proves the bound for every finite vertex type with at least $10$ vertices, which gives the atTop reading below, and collects the uncovered edges as the edge set of a graph rather than as a set of pairs.

depends on sorry what this means
Statement uses: Iff, True, Filter.Eventually, Nat, Sym2, Fin, OfNat.ofNat, LE.le, Set.ncard, Set.ofPred, And, Not, Sym2.IsDiag, Eq, Sym2.mk, Exists, Ne, HDiv.hDiv, HPow.hPow, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_639 : answer(True) ↔
    ∀ᶠ (n : ℕ) in atTop, ∀ C : Sym2 (Fin n) → Fin 2,
      {e : Sym2 (Fin n) | ¬e.IsDiag ∧
        ∀ x y : Fin n, e = s(x, y) →
          ¬∃ z, z ≠ x ∧ z ≠ y ∧ C s(x, z) = C e ∧ C s(y, z) = C e}.ncard ≤ n ^ 2 / 4 := by
…
```