gpt pro 5.2 attempt on Math-833

1) FORMAL RESTATEMENT

Let $m^*$ denote Lebesgue outer measure on $\mathbb R$. For each $x\in\mathbb R$, suppose we are given a set $A_x\subset\mathbb R$ such that: - $A_x$ is bounded, and - $m^*(A_x)<1$.

Call a set $X\subseteq\mathbb R$ independent (for the map $x\mapsto A_x$) if for all distinct $x,y\in X$ we have $x\notin A_y$.

Question 1: Must there exist an infinite independent set $X\subseteq\mathbb R$?

Question 2: If additionally each $A_x$ is closed and has (Lebesgue) measure $m(A_x)<1$, must there exist an independent set of size $3$?

2) QUICK LITERATURE/CONTEXT CHECK

The problem statement reports: (a) Erdős--Hajnal proved existence of arbitrarily large finite independent sets under the assumptions of Question 1; (b) Gladysz proved existence of an independent set of size $2$ under the assumptions of Question 2; (c) Hechler showed Question 1 can fail under CH; (d) Newelski--Pawlikowski--Seredyński proved Question 1 has a positive answer if all $A_x$ are closed.

In what follows I do not use these results as black boxes. Instead I prove a conditional version of the finite independent-set statement under an additional measurability hypothesis.

3) ATTACK PLAN

Proof track: try to build an independent set by a measure/selection argument. The key difficulty is that the set \[E:=\{(u,v)\in\mathbb R^2:\ u\in A_v\} \] need not be measurable in $\mathbb R^2$ without extra hypotheses on the assignment $v\mapsto A_v$.

Construction/disproof track: understand how non-measurability of $E$ could obstruct Fubini-type estimates; identify where additional definability assumptions (Borel/analytic) restore them.

4) WORK

The following lemmas are conditional on a measurability assumption for the relation $E$. This is not part of the problem statement, so these are partial results only.

Assumption (measurability of the membership relation). Assume that the set $E=\{(u,v): u\in A_v\}\subset\mathbb R^2$ is Lebesgue measurable. (For example, this would hold if $(u,v)\mapsto \mathbf 1_{u\in A_v}$ were Borel measurable; the problem statement does not assume this.)

Lemma 501.1 (Independent pair under measurability). Under the measurability assumption on $E$, there exist distinct $x,y\in\mathbb R$ such that $x\notin A_y$ and $y\notin A_x$.

Proof. Fix an interval $I\subset\mathbb R$ of length $L>2$. Consider the measurable sets \[E_I:=E\cap (I\times I),\qquad E_I^T:=\{(u,v)\in I\times I : (v,u)\in E\}. \] By Fubini/Tonelli (applied to the indicator of $E_I$), \[ m_2(E_I)=\int_{v\in I} m_1(\{u\in I: (u,v)\in E\})\,dv =\int_{v\in I} m_1(A_v\cap I)\,dv. \] For each $v$, $m_1(A_v\cap I)\le m^*(A_v)<1$, hence $m_2(E_I)<\int_I 1\,dv = L$. The same argument gives $m_2(E_I^T)<L$. Therefore \[ m_2(E_I\cup E_I^T)\le m_2(E_I)+m_2(E_I^T) < 2L. \] But $m_2(I\times I)=L^2$, and since $L>2$ we have $L^2>2L$. Hence $(I\times I)\setminus (E_I\cup E_I^T)$ has positive measure and is nonempty. Pick $(x,y)$ in this complement with $x\ne y$ (the diagonal has measure $0$). Then $(x,y)\notin E$ implies $x\notin A_y$, and $(x,y)\notin E^T$ implies $y\notin A_x$. Thus $\{x,y\}$ is an independent set. $\square$

Lemma 501.2 (Arbitrarily large finite independent sets under measurability). Under the measurability assumption on $E$, for every integer $m\ge 1$ there exists an independent set $X\subset\mathbb R$ with $\lvert X\rvert\ge m$.

Proof. Fix $m\ge 1$. Choose an interval $I$ of length $L:=8m$. Let $N:=4m$ and sample $N$ i.i.d. points $X_1,\dots,X_N$ uniformly from $I$. Define an undirected graph $G$ on vertex set $\{1,\dots,N\}$ by putting an edge $\{i,j\}$ (with $i\ne j$) if either $X_i\in A_{X_j}$ or $X_j\in A_{X_i}$. If $S\subset\{1,\dots,N\}$ is an independent set in this graph, then by definition for all distinct $i,j\in S$ we have $X_i\notin A_{X_j}$ and $X_j\notin A_{X_i}$; hence $\{X_i:i\in S\}$ is an independent set in the sense of the problem.

We bound the expected number of edges in $G$. For $i\ne j$, \[ \mathbb P[X_i\in A_{X_j}] =\frac{1}{L^2}\,m_2(E\cap (I\times I)) =\frac{1}{L^2}\int_{v\in I} m_1(A_v\cap I)\,dv \le \frac{1}{L^2}\int_I 1\,dv =\frac{1}{L}. \] Similarly $\mathbb P[X_j\in A_{X_i}]\le 1/L$, so by the union bound \[\mathbb P[\{i,j\}\text{ is an edge of }G]\le \frac{2}{L}. \] Let $M$ be the number of edges of $G$. By linearity of expectation, \[ \mathbb E[M] \le \binom{N}{2}\cdot \frac{2}{L} =\frac{N(N-1)}{L} <\frac{N^2}{L}. \] With $N=4m$ and $L=8m$, this gives $\mathbb E[M]<2m$. By Markov's inequality, $\mathbb P[M\le 4m]>0$. Fix an outcome with $M\le 4m$.

For any finite graph with $N$ vertices and $M$ edges, the average degree is $\bar d=2M/N$, and there exists an independent set of size at least $N/(\bar d+1)=N^2/(2M+N)$ (this follows from the Caro--Wei bound or a simple averaging argument). Applying this with $M\le 4m$ and $N=4m$ gives \[ \alpha(G)\ge \frac{N^2}{2M+N}\ge \frac{(4m)^2}{2\cdot (4m) + 4m}=\frac{16m^2}{12m}=\frac{4m}{3}>m. \] Choosing $m$ vertices in such an independent set yields an independent set of size $m$ in $\mathbb R$. $\square$

FAST REALITY CHECK

These lemmas are not about small-$n$ computation but about measure-theoretic structure. A quick sanity example where an infinite independent set exists: if $A_x=[x,x+1/2]$ for all $x$, then any strictly decreasing infinite sequence $x_1>x_2>\cdots$ is independent, since $x_i\notin [x_j,x_j+1/2]$ whenever $i<j$.

5) VERIFICATION

Where measurability is used: Lemmas 501.1--501.2 use Fubini/Tonelli on $\mathbf 1_{E\cap (I\times I)}$. Without measurability of $E$, Fubini-type identities can fail, so these proofs do not resolve the original question.

Quantifiers: The constructed independent sets are finite (size $m$ arbitrary) and depend on the measurability assumption; they do not imply existence of an infinite independent set.

6) FINAL UNRESOLVED

(i) Strongest proved partial result here: If the membership relation $E=\{(u,v):u\in A_v\}$ is Lebesgue measurable, then there exist independent sets of arbitrary finite size (Lemma 501.2), and in particular an independent pair (Lemma 501.1).

(ii) First gap (crisp): Remove the extra hypothesis that $E$ is measurable: prove or disprove in ZFC that for every assignment $x\mapsto A_x$ with each $A_x$ bounded and $m^*(A_x)<1$ there exists an infinite independent set.

(iii) Top 3 next moves: 1. Try to show $E$ is measurable (or sufficiently definable) under natural additional hypotheses (e.g. all $A_x$ Borel/closed) and see whether the argument can be iterated transfinitely to produce an infinite independent set. 2. Attempt a direct transfinite recursion construction of an infinite independent set that avoids needing Fubini on $E$. 3. Search for explicit (ZFC) pathological set-mappings $x\mapsto A_x$ of small outer measure that might block infinite free sets; pin down whether any such construction must use additional set-theory (e.g. CH).

(iv) Minimal counterexample structure: a counterexample in ZFC to Question 1 would likely define a highly non-measurable relation $E\subset\mathbb R^2$ with all vertical sections $E_v=A_v$ of outer measure $<1$, but such that $E\cup E^T$ covers $X\times X$ for every infinite $X\subset\mathbb R$ (forcing an edge in every infinite subset).