570.lean`

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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 570

References:

erdosproblems.com/570
[EFRS93] Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399.
[GoKl94] Goddard, W. and Kleitman, D. J., An upper bound for the Ramsey numbers $r(K_3,G)$. Discrete Math. (1994), 177-182.
[Si91] Sidorenko, A. F., An upper bound on the Ramsey number $r(K_3,G)$ depending only on the size of the graph $G$. J. Graph Theory (1991), 15-17.
[Ja99] Jayawardene, C. J., Ramsey numbers related to small cycles. University of Memphis (1999).
[CFMPP26] Cambie, S., Freschi, A., Morawski, P., Petrova, K. and Pokrovskiy, A., Ramsey number of a cycle versus a graph of a given size. arXiv:2601.10238 (2026).

Imports (1)

FormalConjecturesUtil

theorem Erdos570.erdos_570 Let $k\geq 3$ sorry line 55 source
Source documentation
Let $k\geq 3$. Is it true that, if $m$ is sufficiently large, for any graph $H$ on $m$ edges without isolated vertices, $$R(C_k,H) \leq 2m+\left\lfloor\frac{k-1}{2}\right\rfloor?$$
This was proved for even $k$ by Erdős, Faudree, Rousseau, and Schelp [EFRS93]. This was proved for $k=3$ independently by Goddard and Kleitman [GoKl94] and Sidorenko [Si91]. This was proved for $k=5$ by Jayawardene [Ja99]. Finally it was proved for all odd $k\geq 7$ by Cambie, Freschi, Morawski, Petrova, and Pokrovskiy [CFMPP26].
This problem is #35 in Ramsey Theory in the graphs problem collection.

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Filter.Eventually, Fintype, SimpleGraph, DecidableRel, SimpleGraph.Adj, LT.lt, SimpleGraph.degree, Membership.mem, Set, SimpleGraph.neighborSet, Eq, Set.ncard, Sym2, SimpleGraph.edgeSet, SimpleGraph.graphRamsey and 7 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_570 : answer(True) ↔
    ∀ (k : ℕ) (hk : 3 ≤ k),
      ∀ᶠ (m : ℕ) in atTop,
        ∀ (W : Type) [Fintype W] (H : SimpleGraph W) [DecidableRel H.Adj],
          (∀ v, 0 < H.degree v) →
          H.edgeSet.ncard = m →
          SimpleGraph.graphRamsey (SimpleGraph.cycleGraph k) H ≤ 2 * m + (k - 1) / 2 := by
…
```
Mentions: SimpleGraph.graphRamsey