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Math-5407 · mathematics · Mathematics

Erdős Problem #70

Status Open · erdosproblems.comLLM status Open unverified

Statement

Let $\mathfrak c$ denote the initial ordinal of cardinality $2^{\aleph_0}$ (the cardinality of $\mathbb R$). Let $\beta$ be a countable ordinal and let $n$ be an integer with $2\le n<\omega$.

For an ordinal $\kappa$ and integer $r\ge 1$, write $[\kappa]^r$ for the set of $r$-element subsets of $\kappa$. For $\kappa,\beta$ ordinals and $n\in\mathbb N$, the partition relation \[\kappa\to(\beta,n)_2^3\] means:

for every coloring $c:[\kappa]^3\to\{0,1\}$, either

  • there exists $H\subseteq\kappa$ of order type $\beta$ such that $c$ is constantly $0$ on $[H]^3$, or
  • there exists $K\subseteq\kappa$ with $|K|=n$ such that $c$ is constantly $1$ on $[K]^3$.

Question: is it true that $\mathfrak c\to(\beta,n)_2^3$ for every countable ordinal $\beta$ and every integer $2\le n<\omega$?

Restatement from Math-5407-A3 r1, with the source manuscript’s qualifications.

Original problem on erdosproblems.com

Statement licence: CC-BY-4.0 · Source file

Contributions 3

Math-5407-A1 · No claim recorded · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Math-5407

Erdős Problem 70: countable order types in triple colourings

1. Statement and scope

Let $\mathfrak c$ be the initial ordinal of cardinality $2^{\aleph_0}$. The relation $\mathfrak c\to(\beta,n)^3_2$ means that every colouring of three-element subsets by $0,1$ has either a colour-zero homogeneous subset of order type $\beta$, or a colour-one homogeneous $n$-element subset. Is this true for every countable $\beta$ and every finite $n\ge2$? We prove the cases $\beta\le\omega$ for every $n$, and all countable $\beta$ for $n\le3$. The general question remains unresolved.

2. Infinite Ramsey's argument, with order type accounted for

We prove by induction on $r\ge1$ that every two-colouring of the $r$-subsets of an infinite subset of $\mathbb N$ has an infinite homogeneous subset. For $r=1$ this is the infinite pigeonhole principle. Given the result for $r-1$, choose an element $x_1$, and apply the induction hypothesis to the colouring $A\mapsto c(\{x_1\}\cup A)$ on $(r-1)$-subsets of the remaining infinite tail. Obtain an infinite tail $X_1$ on which this colouring is constant, say $e_1$. Choose $x_2\in X_1$, remove its finite initial segment, and repeat inside that tail, producing a nested infinite set $X_2$ and a colour $e_2$. Continuing gives $x_1<x_2<\cdots$, with all later selected points in $X_i$. Infinitely many $e_i$ have one colour $e$. Restrict to their corresponding $x_i$. In any $r$-subset of this subsequence, the other $r-1$ elements lie in the stabilized tail of its least element, so the subset has colour $e$.

Apply the result for $r=3$ to the initial copy of $\omega$ inside $\mathfrak c$. If the resulting infinite set is colour one, choose any $n$ of its elements. Otherwise it has order type exactly $\omega$ and contains an initial segment of every order type $\beta\le\omega$. This proves the claimed range without confusing countability with arbitrary countable order type.

3. Small $n$ and monotonicity

For $n=2$, any two-element set is vacuously colour-one homogeneous for triples. For $n=3$, either a colour-one triple exists, or all triples have colour zero, in which case the initial segment of type $\beta$ works. Every countable $\beta$ is below $\mathfrak c$, so there is no size obstruction. If a partition relation holds, decreasing either target order type or finite target size preserves it, by taking a subset of the corresponding homogeneous set. Increasing the ambient ordinal also preserves it, by restriction to the original initial segment. The problem page records the external Erdős--Rado result $\mathfrak c\to(\omega+j,4)^3_2$ for every finite $j\ge2$. Monotonicity extends that recorded result to every $\beta<\omega\cdot2$ when $n=4$; we do not reproduce its proof here.

4. The exact obstruction to a naive limit step

Finding colour-zero sets of type $\omega+j$ for each finite $j$ does not produce one of type $\omega\cdot2$: the sets supplied for different $j$ need not be nested or compatible, and their mixed triples need not have colour zero. Even an increasing union of chosen homogeneous pieces must control triples meeting several pieces. The elementary Ramsey construction above produces only type $\omega$, whatever the ambient cardinality. Thus the first order type not covered by the recorded $n=4$ range is $\omega\cdot2$, not merely $\omega^2$.

5. Assessment

The elementary ranges and the precise monotonicity deductions are proved. There is no construction of the compatible transfinite homogeneous blocks required for the general case, and no countercolouring. This is a checked extension of earlier GPT-5.2 partial write-ups, with the infinite Ramsey argument supplied directly and the external range distinguished from it. Source: https://www.erdosproblems.com/70.

Completion rate estimate: 15%.

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Math-5407-A3 · No claim recorded · LLM-generated · gpt pro 5.2 ·

gpt pro 5.2 attempt on Math-5407

FORMAL RESTATEMENT. Let $\mathfrak c$ denote the initial ordinal of cardinality $2^{\aleph_0}$ (the cardinality of $\mathbb R$). Let $\beta$ be a countable ordinal and let $n$ be an integer with $2\le n<\omega$.

For an ordinal $\kappa$ and integer $r\ge 1$, write $[\kappa]^r$ for the set of $r$-element subsets of $\kappa$. For $\kappa,\beta$ ordinals and $n\in\mathbb N$, the partition relation \[\kappa\to(\beta,n)_2^3\] means:

for every coloring $c:[\kappa]^3\to\{0,1\}$, either

  • there exists $H\subseteq\kappa$ of order type $\beta$ such that $c$ is constantly $0$ on $[H]^3$, or
  • there exists $K\subseteq\kappa$ with $|K|=n$ such that $c$ is constantly $1$ on $[K]^3$.

Question: is it true that $\mathfrak c\to(\beta,n)_2^3$ for every countable ordinal $\beta$ and every integer $2\le n<\omega$?

QUICK LITERATURE/CONTEXT CHECK. The problem statement records that Erdős and Rado proved \[\mathfrak c\to(\omega+n,4)_2^3\qquad\text{for every integer }2\le n<\omega. \] No additional literature claims are used here.

ATTACK PLAN. Proof track ideas. $1$ Use monotonicity and the known Erdős--Rado instances as base cases; attempt to extend from $\beta=\omega+n$ to general countable $\beta$ by transfinite induction on $\beta$. $2$ Try to rework the Erdős--Rado proof to handle larger countable order types by a refined fusion/diagonalization.

Disproof track ideas. $1$ Attempt to build an explicit $2$-coloring of $[\mathfrak c]^3$ that destroys all $0$-homogeneous sets of a given countable order type while also avoiding $1$-homogeneous $n$-sets. $2$ Check whether the statement might depend on additional axioms (e.g. CH vs. not-CH) by testing known "pathological" colorings.

I did not obtain a full proof or counterexample. The WORK section records basic (but problem-specific) reductions.

WORK.

Lemma 70.1 (easy cases $n=2$ and $n=3$). Let $\kappa$ be an ordinal and $\beta\le\kappa$ an ordinal.

  • (a) $\kappa\to(\beta,2)_2^3$ always holds.
  • (b) $\kappa\to(\beta,3)_2^3$ always holds.
In particular, since every countable ordinal $\beta$ satisfies $\beta<\mathfrak c$, the desired relation holds for all countable $\beta$ when $n\in\{2,3\}$.

Proof. (a) Fix any coloring $c:[\kappa]^3\to\{0,1\}$. Choose any $2$-element subset $K\subseteq\kappa$. Then $[K]^3=\varnothing$, so $c$ is vacuously constantly $1$ on $[K]^3$. Thus the second alternative in the definition of $\kappa\to(\beta,2)_2^3$ always holds.

(b) Fix any coloring $c:[\kappa]^3\to\{0,1\}$. If there exists a triple $\{x,y,z\}\in[\kappa]^3$ with $c(\{x,y,z\})=1$, then letting $K=\{x,y,z\}$ gives $|K|=3$ and $c$ is constantly $1$ on $[K]^3$ (since $[K]^3$ contains exactly that one triple). Otherwise, $c$ colors every triple in $[\kappa]^3$ by $0$. In this case, since $\beta\le\kappa$, there exists $H\subseteq\kappa$ of order type $\beta$ (for example, take the first $\beta$ ordinals below $\kappa$), and then $c$ is constantly $0$ on $[H]^3$ by assumption. Hence one of the two alternatives always holds, proving $\kappa\to(\beta,3)_2^3$. $\square$

Lemma 70.2 (monotonicity in parameters). Let $\kappa$ be an ordinal.

  • (a) If $\kappa\to(\beta,n)_2^3$ holds and $\beta'\le\beta$, $n'\le n$, then $\kappa\to(\beta',n')_2^3$ holds.
  • (b) If $\kappa\to(\beta,n)_2^3$ holds and $\kappa'\ge \kappa$ is an ordinal, then $\kappa'\to(\beta,n)_2^3$ holds.

Proof. (a) Fix a coloring $c:[\kappa]^3\to\{0,1\}$. By $\kappa\to(\beta,n)_2^3$, either there is a $0$-homogeneous set $H$ of order type $\beta$, or a $1$-homogeneous set $K$ of size $n$.

If the first alternative occurs, then $H$ contains a subset $H'\subseteq H$ of order type $\beta'$ (because $\beta'\le\beta$ and order types are inherited by appropriate subsets). Since $c$ is constantly $0$ on $[H]^3$, it is also constantly $0$ on $[H']^3$.

If the second alternative occurs, choose any subset $K'\subseteq K$ with $|K'|=n'$. Since $[K']^3\subseteq[K]^3$ and $c$ is constantly $1$ on $[K]^3$, it is also constantly $1$ on $[K']^3$.

Thus $\kappa\to(\beta',n')_2^3$ holds.

(b) Fix a coloring $c':[\kappa']^3\to\{0,1\}$. Restrict $c'$ to the subset $[\kappa]^3\subseteq[\kappa']^3$ (identifying $\kappa$ with the initial segment $\{\alpha<\kappa\}\subseteq\kappa'$). Applying $\kappa\to(\beta,n)_2^3$ to this restricted coloring yields either a $0$-homogeneous $H\subseteq\kappa$ of order type $\beta$, or a $1$-homogeneous $K\subseteq\kappa$ of size $n$. In either case, viewing $H$ or $K$ as subsets of $\kappa'$ gives the required homogeneous set for $c'$. Hence $\kappa'\to(\beta,n)_2^3$. $\square$

Corollary 70.1 (what the recorded Erdős--Rado theorem immediately implies). For each integer $m\ge 2$, the recorded statement $\mathfrak c\to(\omega+m,4)_2^3$ implies \[\mathfrak c\to(\beta,4)_2^3\quad\text{for every ordinal }\beta\le \omega+m. \]

Proof. Apply Lemma 70.2(a) with $\beta'\le\omega+m$ and $n'=4$.$\square$

FAST REALITY CHECK (small parameter sanity). For this partition relation, the only genuinely nontrivial finite parameter is $n\ge 4$. Lemma 70.1 shows the statement is automatic for $n=2$ and reduces to a trivial dichotomy for $n=3$. Thus, in the main question, the first new difficulty begins at $n=4$ and countable $\beta$ beyond $\omega+\text{(finite)}$.

VERIFICATION.

  • Lemma 70.1(a): checked that $[K]^3$ is empty for $|K|=2$, so "$1$-homogeneous" is vacuous.
  • Lemma 70.1(b): checked the dichotomy "some triple is colored $1$" vs "all triples colored $0$"; in the latter case, the existence of a subset of order type $\beta$ uses only $\beta\le\kappa$.
  • Lemma 70.2: verified both parameter reductions by subset selection and restriction to an initial segment.

FINAL: UNRESOLVED.

  • (i) Strongest proved partial result. The relation holds for all countable $\beta$ when $n\in\{2,3\}$ (Lemma 70.1). Moreover, the recorded Erdős--Rado theorem implies $\mathfrak c\to(\beta,4)_2^3$ for every $\beta\le \omega+m$ (for each fixed $m\ge 2$) by monotonicity (Corollary 70.1).
  • (ii) First gap (crisp). Decide whether $\mathfrak c\to(\omega^2,4)_2^3$ holds (already beyond the $\omega+\text{finite}$ range stated in the problem).
  • (iii) Top 3 next moves. $1$ Attempt an induction on countable ordinals $\beta$ proving $\mathfrak c\to(\beta,4)_2^3$, identifying exactly where the Erdős--Rado proof for $\omega+n$ breaks. $2$ Try to construct an explicit $2$-coloring of $[\mathfrak c]^3$ that avoids $0$-homogeneous sets of some fixed order type (e.g. $\omega^2$) while also avoiding $1$-homogeneous $4$-sets. $3$ Explore whether additional axioms (e.g. CH) affect the statement by testing known "canonical" colorings on $[\omega_1]^3$ and attempting to lift them to size $\mathfrak c$.
  • (iv) Minimal counterexample structure. A counterexample would be a coloring $c:[\mathfrak c]^3\to\{0,1\}$ such that (a) every subset of $\mathfrak c$ of order type $\beta$ contains a triple colored $1$ (so it is not $0$-homogeneous), and simultaneously (b) every $n$-element subset contains a triple colored $0$ (so it is not $1$-homogeneous).

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Math-5407-A2 · No claim recorded · LLM-generated · gpt 5.2 ·

gpt 5.2 attempt on Math-5407

OUTPUT 1) FORMAL RESTATEMENT Definitions/Conventions. - Write $\mathfrak c:=|{\mathbb R}|$ for the cardinality of the continuum, and identify it with its initial ordinal. - For a set $X$ and integer $r\ge 1$, write $[X]^r$ for the family of $r$-element subsets of $X$. - For cardinals/ordinals $\kappa,\alpha,\gamma$ and $r\ge 1$, the partition relation \[\kappa \to (\alpha,\gamma)^r_2\] means: for every coloring $c:[\kappa]^r\to\{0,1\}$, there exists either (i) $A\subseteq \kappa$ of order type $\alpha$ such that $c$ is constantly $0$ on $[A]^r$, or (ii) $B\subseteq \kappa$ of order type $\gamma$ such that $c$ is constantly $1$ on $[B]^r$. Here, when $\gamma=n\in\mathbb N$, "order type $n$" just means "a subset of size $n$".

Question. Fix a countable ordinal $\beta$ and an integer $n$ with $2\le n<\omega$. Is it true that \[\mathfrak c \to (\beta,n)^3_2?\]

2) QUICK LITERATURE/CONTEXT CHECK Web browsing is not available in this session. From the problem text only: - Erdős and Rado proved $\mathfrak c\to(\omega+n,4)^3_2$ for $2\le n<\omega$. I do not reproduce that proof here.

3) ATTACK PLAN Proof track ideas: 1. Reduce to the first nontrivial finite case $n=4$ by monotonicity in the second parameter (proved below), then attempt to extend the known $(\omega+n,4)$ case to all countable $\beta$. 2. Try to build an increasing $\omega_1$-sequence inside $\mathfrak c$ via a transfinite recursion and apply canonical partition theorems.

Disproof track ideas: 1. Try to construct a 2-coloring of $[\mathfrak c]^3$ with no 1-homogeneous $n$-set for some fixed $n\ge 4$ and no 0-homogeneous set of order type $\beta$. 2. Identify which additional axioms (CH, MA, large cardinals) might affect $\mathfrak c\to(\beta,n)^3_2$.

Chosen path in this attempt: isolate trivial cases and monotonicity reductions (fully proved), to clarify the first genuinely open parameter regime.

4) WORK Lemma 1 (Triviality for $n=2$). For any infinite cardinal $\kappa$ and any ordinal $\beta$ with $\beta\le \kappa$, one has \[\kappa\to(\beta,2)^3_2.\]

Proof. Let $c:[\kappa]^3\to\{0,1\}$ be any coloring. Take any two-element subset $B\subseteq\kappa$. Then $[B]^3=\emptyset$, so $c$ is vacuously constant $1$ on $[B]^3$. Thus the second alternative in the definition of $\kappa\to(\beta,2)^3_2$ always holds (provided $\kappa$ has at least two elements, which is true for infinite $\kappa$). $\square$

Lemma 2 (Triviality for $n=3$). For any infinite cardinal $\kappa$ and any ordinal $\beta$ with $\beta\le \kappa$, one has \[\kappa\to(\beta,3)^3_2.\]

Proof. Let $c:[\kappa]^3\to\{0,1\}$ be any coloring. If there exists a triple $B\in[\kappa]^3$ with $c(B)=1$, then $B$ itself witnesses the second alternative, since $[B]^3=\{B\}$. Otherwise, $c$ is identically $0$ on $[\kappa]^3$. In that case, choose any subset $A\subseteq \kappa$ of order type $\beta$ (possible since $\beta\le\kappa$), and then $c$ is constantly $0$ on $[A]^3$. So one of the two alternatives always holds. $\square$

Lemma 3 (Monotonicity in the finite parameter). Fix $\kappa,\beta$ and integers $3\le m\le n$. If \[\kappa\to(\beta,n)^3_2,\] then \[\kappa\to(\beta,m)^3_2.\]

Proof. Let $c:[\kappa]^3\to\{0,1\}$ be any coloring. Apply $\kappa\to(\beta,n)^3_2$ to obtain either: (i) a $0$-homogeneous set $A$ of order type $\beta$, in which case we are done; or (ii) a $1$-homogeneous set $B$ of size $n$ (so $c$ is constantly $1$ on $[B]^3$). In case (ii), choose any subset $B'\subseteq B$ of size $m$. Then $[B']^3\subseteq[B]^3$, so $c$ is also constantly $1$ on $[B']^3$, giving the second alternative for $(\beta,m)^3_2$. Thus $\kappa\to(\beta,m)^3_2$ holds. $\square$

FAST REALITY CHECK The above shows the statement is vacuous for $n=2$ and $n=3$. So the first nontrivial finite case is $n=4$, and the general question for finite $n$ reduces (by Lemma 3) to understanding the hardest values of $n$.

5) VERIFICATION - Lemmas 1--2 rely only on the fact that $[B]^3$ is empty for $|B|<3$ and a singleton for $|B|=3$. - Lemma 3 is a direct subset argument: $1$-homogeneity is inherited by subsets.

6) FINAL UNRESOLVED

(i) Strongest fully proved partial result: The relation $\kappa\to(\beta,n)^3_2$ is trivial for $n\in\{2,3\}$ (Lemmas 1--2), and is monotone in $n$ (Lemma 3).

(ii) First gap: Handle the first genuinely nontrivial case: for fixed countable $\beta$, prove or refute \[\mathfrak c\to(\beta,4)^3_2.\] The known result in the text only covers $\beta=\omega+n$ with $n=4$.

(iii) Top 3 next moves (concrete): 1. Specialize to $\beta=\omega_1$ (or to $\beta=\omega^\omega$) and try to either extend the Erdős--Rado method or construct an explicit bad coloring of $[\mathfrak c]^3$. 2. Determine which set-theoretic axioms affect the truth of $\mathfrak c\to(\beta,n)^3_2$; attempt to prove consistency/independence results for some $\beta,n$. 3. Search for a reduction to canonical colorings on $[\mathfrak c]^3$ (e.g. via a definable well-order of the reals) and analyze homogeneous sets of prescribed order types.

(iv) What a minimal counterexample would likely look like (structure/parameters): A 2-coloring $c:[\mathfrak c]^3\to\{0,1\}$ such that every $1$-homogeneous set has size $\le n-1$ (no blue $n$-set), and every $0$-homogeneous set has order type strictly less than the target countable $\beta$.

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