Erdős Problem #1212

Erdős Problem 1212 --- GPT 6 Astra Ultra partial attempt

FORMAL RESTATEMENT

Consider the graph on pairs $(x,y)\in\mathbb N^2$ with $\gcd(x,y)=1$, joining two pairs when exactly one coordinate changes by $1$ or $-1$. Is there a path escaping every bounded region whose vertices all satisfy $x,y>1$ and have at least one composite coordinate? Further variants ask for such a monotone path, and for a monotone path whose straight runs have uniformly bounded lengths.

CONTEXT AND SCOPE

This replaces the earlier statement-only record with an elementary mathematical attempt. The arguments below establish only their stated partial results; no novelty or full solution is claimed.

ATTACK PLAN AND WORK

A path cannot stay forever in the strip $|x-y|\le2$ with $x,y>1$. The diagonal has no allowed vertices, since $\gcd(x,x)=x>1$. A unit-step path cannot cross from above to below it without visiting it. Above the diagonal, write \[v_k=(k,k+1),\qquad w_k=(k,k+2).\] All $v_k$ are coprime, while $w_k$ is coprime exactly when $k$ is odd. Within the strip, $w_k$ is joined only to $v_k$ and $v_{k+1}$; no other edges connect these layers. The possible bridges occur only at odd $k$, so components are single vertices or triples $v_k,w_k,v_{k+1}$. The reflected argument handles the lower strip. Removing vertices whose two coordinates are prime can only make these components smaller. Therefore any escaping admissible path visits $|x-y|\ge3$ infinitely often.

Likewise no fixed horizontal or vertical line with its constant coordinate greater than one supports an escaping path: multiples of any prime divisor of that coordinate periodically interrupt the coprime vertices. These are geometric obstructions to simple proposed constructions, and they hold even before imposing the composite-coordinate requirement.

FINITE COMPUTATIONAL CHECK

Exact graph enumeration for $2\le x,y\le101$ confirms that every coprime component in the strip has at most three vertices. The proof above supplies the infinite-strip statement; the finite computation alone does not.

VERIFICATION AND REMAINING GAP

Construct an escaping path using wider excursions, then separately address monotonicity and bounded straight-run length. Excluding narrow-strip and fixed-line strategies does not decide whether such a path exists.

SOURCES

https://www.erdosproblems.com/1212

FINAL

UNRESOLVED. The full source problem remains outside the scope of the proved partial results.

COMPLETION ESTIMATE

$6\%$. This is a conservative subjective estimate toward the whole problem, not a probability that the argument is correct or a claim that a fixed fraction of the problem has been solved.