Erdős Problem #1206

Erdős Problem 1206 --- GPT 6 Astra Ultra partial attempt

FORMAL RESTATEMENT

Is there a constant $c>0$ such that for every sufficiently large $N$, the first $N$ positive cubes contain a Sidon subset of at least $cN$ elements? Is there an infinite set $A\subseteq\mathbb N$ of positive density for which $\{a^3:a\in A\}$ is Sidon? Here Sidon means uniqueness of every two-term sum up to exchanging its summands.

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.

The page leaves the convention for positive density implicit. Its remarks also raise higher-power analogues; these are not treated as already solved.

ATTACK PLAN AND WORK

A greedy argument works for every $N$-element set $X$ of distinct real numbers, and hence for the first $N$ cubes. Suppose a Sidon subset $S$ has size $m$. A new element $x$ can violate the property only through a collision $x+a=b+c$ with $a,b,c\in S$, or $2x=a+b$ with $a,b\in S$, or by already belonging to $S$. Equalities $x+a=x+b$ force $a=b$ and cause no new violation. Thus at most $m^3+m^2+m$ real values are forbidden.

If $S$ is maximal within $X$, all of $X$ must be forbidden, giving \[N\le m^3+m^2+m\le3m^3\quad(m\ge1). \] Therefore every such $X$ contains a Sidon subset with at least $(N/3)^{1/3}$ elements. This proof handles repeated summands through the $2x$ condition.

An infinite example is $A=\{2^j:j\ge0\}$: its cubes are $8^j$, and uniqueness of their two-term sums follows from base-eight digits, including a single digit two when a summand repeats. But $A$ has density zero. Neither this infinite construction nor the finite cube-root bound supplies the positive proportion or positive-density conclusion asked for.

VERIFICATION AND REMAINING GAP

Use special arithmetic of cubes to improve a generic cube-root-sized selection to a linear-sized one, and make the selections compatible across scales for the infinite positive-density version.

SOURCES

https://www.erdosproblems.com/1206

FINAL

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

COMPLETION ESTIMATE

$8\%$. 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.