gpt pro 5.2 attempt on Math-5793

Problem 650 (distinct multiples in any interval of length $2N$)

FORMAL RESTATEMENT

Let $m\ge 1$. Define $f(m)$ to be the largest integer with the following property:

For every integer $N\ge 1$ and every set $A\subseteq\{1,\dots,N\}$ with $|A|=m$, every interval $I\subseteq[1,\infty)$ of length $2N$ contains at least $f(m)$ distinct integers $b_1,\dots,b_{f(m)}\in I$ such that each $b_i$ is divisible by a different element $a_i\in A$.

Equivalently: for each such $(N,A,I)$, in the bipartite graph with left part $A$ and right part $I\cap\mathbb Z$, where $a\sim b$ iff $a\mid b$, the maximum matching has size at least $f(m)$.

The question is to estimate $f(m)$ as $m\to\infty$, and in particular whether \[ f(m)\ll m^{1/2}. \]

QUICK LITERATURE/CONTEXT CHECK (browsing)

The problem statement on the Erdős Problems website records that Erdős and Sárányi $1959$ proved a lower bound of order \[f(m)\gg m^{1/2}. \] I did not find, in a brief search, a matching $O(m^{1/2})$ upper bound or a resolution of the posed question.

ATTACK PLAN

1. (Definitions/normalization) Recast as a matching problem in a bipartite graph.
2. (Try small parameters) Compute/guess $f(m)$ for $m\le 4$ to see the behaviour.
3. (Known theorems) Consider Hall's theorem, Kőnig's theorem (for bipartite graphs), and divisor-counting bounds.
4. (Probabilistic attempt) Randomly sample integers in $I$ and estimate how many distinct divisors from $A$ one can "capture".
5. (Construction attempt for upper bounds) Try to build $(N,A,I)$ with small maximum matching, to show $f(m)$ cannot be too large.
6. (Contradiction attempt for lower bounds) Assume maximum matching $<r$ and use Hall to force a structured obstruction; attempt to rule this out beyond $r\approx c\sqrt m$.

WORK

Step 1: Trivial universal bounds (proved)

Step 2: Matching reformulation via Hall

Fix $(N,A,I)$. Let $B:=I\cap\mathbb Z$ so $|B|=2N$. A matching of size $r$ corresponds to choosing distinct $a_1,\dots,a_r\in A$ and distinct $b_1,\dots,b_r\in B$ with $a_i\mid b_i$.

By Hall's theorem, the maximum matching size is $<r$ if and only if there exists a subset $S\subseteq A$ with \[ |N(S)|<|S|, \] where $N(S)\subseteq B$ is the set of all integers in $B$ divisible by at least one element of $S$.

A route to proving $f(m)\gg \sqrt m$ would be to show that such a Hall obstruction cannot occur with $|S|$ much larger than $\sqrt m$.

Step 3: Incomplete attempts

I attempted two approaches:

• Divisor-multiplicity averaging: lower bound $|N(S)|$ in terms of $\sum_{a\in S} (2N/a)$, and upper bound the number of $a\in S$ that can divide a fixed $b\in B$. This runs into the difficulty that $b$ can be very large (the interval is not bounded above), so $b$ can have extremely many divisors $\le N$.

• Poset/chain--antichain ideas: try to find a large subfamily of $A$ with a controlled divisibility structure (e.g. a long chain), for which distinct representatives can be chosen. I did not obtain a clean universal guarantee from this method without invoking deeper results.

VERIFICATION

Only the trivial inequalities are proved here; they are straightforward.

FINAL

UNRESOLVED.

Strongest fully proved partial result in this write-up:

The universal bounds $1\le f(m)\le m$.

The nontrivial lower bound $f(m)\gg \sqrt m$ (Erdős--Sárányi, 1959) and the questioned upper bound $f(m)\ll \sqrt m$ remain outside what I was able to prove from scratch here.

COMPLETION ESTIMATE

$0.10$.