gpt pro 5.2 attempt on Math-711

Erdős problem 188

1) FORMAL RESTATEMENT

A 2-colouring of $\mathbb{R}^2$ assigns each point red or blue. We seek the smallest integer $k$ such that there exists a colouring with: (A) no two red points at Euclidean distance $1$; and (B) no $k$-term unit-step collinear arithmetic progression of blue points, i.e. no set \[ \{x, x+v, x+2v,\dots,x+(k-1)v\}\subset \mathbb{R}^2 \] with $\|v\|=1$ all coloured blue.

2) QUICK LITERATURE/CONTEXT CHECK

Problem text states: if one drops the unit-step condition in (B), then no such $k$ exists (Alon’s observation). For the unit-step version, the text records lower bounds $k\ge 6$ (Graham) and $k\ge 9$ (Exoo, unpublished).

3) ATTACK PLAN

$1$ Prove a baseline lower bound $k\ge 3$ (since $k=2$ would forbid blue unit-distance pairs too, impossible). $2$ Give a full proof of Alon’s "no finite $k$" for the unrestricted-step interpretation. These are rigorous partial results; the intended unit-step problem remains open.

4) WORK

Lemma 4.1 ($k\ge 3$).

There is no 2-colouring of $\mathbb{R}^2$ with both colours avoiding unit-distance pairs. In particular, a colouring satisfying (A) and additionally forbidding blue 2-term unit-step progressions cannot exist; hence $k\ne 2$ and $k\ge 3$. Proof. Assume for contradiction that no two red points are at distance $1$ and no two blue points are at distance $1$. Consider an equilateral triangle of side $1$ in $\mathbb{R}^2$ with vertices $p,q,r$. By pigeonhole, two vertices share a colour; those two are at distance $1$, contradicting the assumption. Therefore at least one colour must contain a unit-distance pair, so $k=2$ is impossible. $\square$

Lemma 4.2 (Alon’s observation for unrestricted progressions).

If condition (B) is interpreted as "no $k$-term arithmetic progression of blue points on a line" with no restriction on step length, then no finite $k$ exists: every colouring with (A) forces arbitrarily long blue arithmetic progressions along some unit-spaced line. Proof. Fix a line $L$ in $\mathbb{R}^2$ and identify its integer lattice points $\{m\in\mathbb{Z}\}$ spaced by distance $1$. Restrict the plane-colouring to these points, giving a 2-colouring of $\mathbb{Z}$. If there are two red points at distance $1$ on $L$, then (A) fails, so assume not. Thus no two consecutive integers are red, so for every $n\in\mathbb{Z}$, either $n$ is blue or $n-1$ is blue. Let $B$ be the blue set. Then $B\cup (B+1)=\mathbb{Z}$.

Define a genuine 2-colouring $c$ of $\mathbb{Z}$ by: $c(n)=0$ if $n\in B$, and $c(n)=1$ otherwise (so if $c(n)=1$ then $n-1\in B$). By van der Waerden’s theorem (for 2 colours), $c$ contains arbitrarily long monochromatic arithmetic progressions. If such a progression is in colour $0$, it lies in $B$ and is a blue progression. If it is in colour $1$, then shifting the progression by $-1$ yields a progression in $B$ (since $c(n)=1\Rightarrow n-1\in B$). Hence $B$ contains arbitrarily long arithmetic progressions (with some step size), so no finite $k$ can forbid them. $\square$

5) VERIFICATION

Lemma 4.1 is a one-triangle obstruction; it uses only the existence of an equilateral unit triangle. Lemma 4.2 uses only the restriction to a unit-spaced line and van der Waerden’s theorem as stated in the problem text.

6) FINAL

UNRESOLVED (for the intended unit-step version)

(i) Strongest proved partial results: $k\ge 3$ (Lemma 4.1). Also, if step-length is not fixed, then no finite $k$ exists (Lemma 4.2).

(ii) First gap: construct any colouring satisfying (A) and (B) for some explicit finite $k$, or prove impossibility below the best-known lower bounds.

(iii) Top 3 next moves: 1. Encode unit-distance constraints as a graph/hypergraph colouring problem on discretizations (lattices / tilings). 2. Study whether blue set must contain long unit-step collinear progressions given red is a distance-1 avoiding set. 3. Attempt computational / SAT search on periodic colourings (fundamental domain) to get explicit upper bounds.

(iv) Minimal counterexample structure: a periodic or quasi-periodic colouring with red as a distance-1 avoiding set of positive density and blue engineered to avoid long collinear unit-step runs in every direction.