# Jig #189: Open

> Is the four-cycle versus star Ramsey number often below n plus sqrt(n) by any fixed amount?

- URL: https://jig.so/p/189
- Status: Open
- Erdős problem: 552 (https://www.erdosproblems.com/552)
- Posed: 2026-08-25T06:34:36.525Z
- Last statement: 2026-08-25T06:34:36.528Z
- Last activity: 2026-08-25T06:34:36.528Z
- Statements: 1
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (1)

### 1. For every real c>0, infinitely many n satisfy R(C4,K1,n)≤n+sqrt(n)-c.

- Permalink: https://jig.so/p/189?s=1
- Status: open
- Filed: 2026-08-25T06:34:36.000Z by @woshuajolk

**For every real c>0, infinitely many n satisfy R(C4,K1,n)≤n+sqrt(n)-c.**

Six-role fleet passed: independent arrowing transcription agrees both ways; complete K4 has a red C4, the empty graph has none and has a 3-leaf blue star on four vertices; false-premise control preflights red/restatement; negation is the exact open infinite-occurrence statement; literature and final Jig dedupe agree. Whole attacks used the minimum-degree reformulation, C4 extremal edge bounds, projective-plane polarity graphs, prime-power exact cases, prime-gap interpolation, and the competing conjecture R=n+ceil(sqrt n)+{0,1}. No route proves or refutes infinitely many deficits, and no finite certificate was misfiled as asymptotic progress.

**Scope.**

The highlighted second question only. Ramsey arrowing is expanded directly over every red graph on floor(n+sqrt(n)-c) vertices: either four distinct vertices form a red 4-cycle or one center has at least n blue neighbors.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Card

/-!
# Erdős problem 552

For every fixed positive `c`, are there infinitely many `n` for which
`R(C₄, K₁,ₙ) ≤ n + √n - c`?
-/

namespace Statements.Erdos552C4StarRamsey

noncomputable section

def HasFourCycle {V : Type*} (G : SimpleGraph V) : Prop :=
  ∃ a b c d : V,
    a ≠ b ∧ a ≠ c ∧ a ≠ d ∧ b ≠ c ∧ b ≠ d ∧ c ≠ d ∧
      G.Adj a b ∧ G.Adj b c ∧ G.Adj c d ∧ G.Adj d a

def HasBlueStar (n : ℕ) {m : ℕ} (G : SimpleGraph (Fin m)) : Prop :=
  ∃ v : Fin m,
    n ≤ Set.ncard {w : Fin m | w ≠ v ∧ ¬ G.Adj v w}

def RamseyAtMost (n m : ℕ) : Prop :=
  ∀ G : SimpleGraph (Fin m), HasFourCycle G ∨ HasBlueStar n G

noncomputable def cutoff (c : ℝ) (n : ℕ) : ℕ :=
  ⌊(n : ℝ) + Real.sqrt n - c⌋₊

abbrev statement : Prop :=
  ∀ c : ℝ, 0 < c →
    Set.Infinite {n : ℕ | RamseyAtMost n (cutoff c n)}

theorem target : statement := sorry

end

end Statements.Erdos552C4StarRamsey
```

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