Erdős Problem #189

Erdős Problem #189

1. FORMAL RESTATEMENT

Must every finite colouring of the plane have a colour class containing the four vertices of a rectangle of every prescribed positive area?

2. LITERATURE AND SCOPE

The answer is negative. This is an independently checked reconstruction of Vjekoslav Kovač's explicit 25-colour construction, using the quadratic invariant in his author-supplied proof blueprint. It is a known counterexample, not a claim of novelty. Rectangles may have any orientation.

3. CONSTRUCTION

Identify the plane with $\mathbb C$. For a real $t$, write $\{t\}=t-\lfloor t\rfloor$. Give $z$ the colour \[ c(z)=\left(\left\lfloor5\left\{\frac3{10}\Re(z^2)\right\}\right\rfloor, \left\lfloor5\left\{\frac3{10}\Im(z^2)\right\}\right\rfloor\right)\in\{0,1,2,3,4\}^2. \] The half-open bins make this a well-defined colouring, including on their boundaries.

4. WHY IT WORKS

Suppose that $z,z+u,z+u+v,z+v$ have a common colour, and put \[ I=z^2-(z+u)^2+(z+u+v)^2-(z+v)^2=2uv. \] In each real coordinate of $3z^2/10$, the four fractional parts lie in the same interval of length $1/5$. Their alternating sum therefore lies strictly between $-2/5$ and $2/5$. The integer parts contribute a Gaussian integer. Consequently \[ I=\frac{10}{3}(a+ib+\xi+i\eta),\qquad a,b\in\mathbb Z,\quad |\xi|,|\eta|<\frac25. \] If $a=b=0$, then $|I|<(10/3)(2\sqrt2/5)=4\sqrt2/3<2$. If at least one of $a,b$ is nonzero, the absolute value of the corresponding coordinate of $a+ib+\xi+i\eta$ is greater than $3/5$, so $|I|>2$. Thus a monochromatic parallelogram cannot have $|I|=2$.

For a rectangle, its area is $|u||v|$, and $|I|=2|u||v|$. An area-one rectangle would therefore have $|I|=2$, a contradiction. In fact the same argument excludes any parallelogram whose consecutive side lengths have product one. It does not exclude all area-one parallelograms, whose angle need not be a right angle.

For any fixed $A>0$, the colouring $z\mapsto c(z/\sqrt A)$ excludes rectangles of area $A$. The original question already fails at $A=1$.

5. VERIFICATION AND FINAL

SOLVED: negative answer, by reconstruction of a known proof. The strict open error intervals are essential: a noncentral square has coordinate magnitude strictly greater than $3/5$. The calculation covers rotations and translations and uses neither a generic-position assumption nor a finite numerical search. There is no remaining proof gap and no external theorem beyond elementary complex arithmetic.

6. SOURCES

Kovač, Coloring and density theorems for configurations of a given volume, https://arxiv.org/abs/2309.09973; author's proof blueprint, https://web.math.pmf.unizg.hr/~vjekovac/EP/EP189/Erdos189_blueprint.tex. Current statement: https://www.erdosproblems.com/189.

COMPLETION ESTIMATE

COMPLETION: 100%