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Claim. For every cardinal $\mu$ there is a family $\mathcal A$ of countably infinite sets with $\lvert A\cap B\rvert\neq 2$ for all distinct $A,B\in\mathcal A$ such that every coloring of $\bigcup\mathcal A$ with $\mu$ colors makes some member of $\mathcal A$ monochromatic. Hence the cardinal asked for in Problem 603, the least number of colors that always suffices, does not exist: no bound holds for all such families. This is Theorem 1 and Corollary 3 of the note "A note on Erdős Problem #603", dated 2026-04-21 and posted on the site's thread the same day by Przemek Chojecki, who writes that GPT-5.4 Pro produced the argument; the note itself carries no author line.

Submission note. Posted to the site's forum by Przemek Chojecki on 21 April 2026:

GPT-5.4 Pro can produce a counterexample in the current wording of the problem too, and also give an answer for another wording. Here's a short note.

(The site has been updated to address this comment.)

Argument. The note derives the family from a partition relation. For an infinite cardinal $\mu$ let $\kappa=(2^\mu)^+$; the Erdős--Rado theorem gives $\kappa\to(\mu^+)^2_\mu$, so every coloring of the pairs of $\kappa$ with $\mu$ colors has a monochromatic set of size $\mu^+$, in particular a countably infinite one (for finite $\mu$, Ramsey's theorem gives the same with $\kappa=\omega$). Take the ground set $V=[\kappa]^2$ and, for each countably infinite $X\subseteq\kappa$, the set $A_X=[X]^2$ of its pairs; in graph terms, $V$ is the edge set of the complete graph on $\kappa$ and $A_X$ the edge set of the complete subgraph on $X$. Two such sets meet in $[X\cap Y]^2$, whose size is $\binom n2$ for a finite $n$, so one of $0,1,3,6,\dots$, or countably infinite, and never $2$. A coloring of $V$ with $\mu$ colors is a coloring of the pairs of $\kappa$, so some $A_H$ is monochromatic. The argument is short enough that the site's commentary restates it; nothing on this page is independently reviewed by this project.

Also in the note. If $(A_i)$ is read as a countable sequence, the intersection condition is irrelevant and two colors always suffice (Proposition 6), so the least number of colors is exactly $2$ in that reading. A second note of 2026-04-22, written after a thread comment suggested quantitative versions, gives bounds on the size of the ground set and of the family in terms of the number of colors; it is linked above, and its bounds are not recorded here.

Two formulations. Erdős's own wording, in Problem 12 of Erdős 1987 (printed p. 227), is a yes-or-no question: "Is there a bound on the chromatic number of such a family?" The site reformulates it as a request to find the smallest cardinal $C$ that always suffices. The theorem answers Erdős's question in the negative, and under the site's formulation it determines that the cardinal asked for does not exist. The claim value is answered, the value for a find question whose answer is neither a proof nor a disproof of a stated assertion; the site's label is SOLVED. The thread noted the difference between the two wordings (post 5675) and the extension of the argument to a conjecture of Komjáth, and the curator remarked (post 7373) that so direct a corollary of the Erdős--Rado theorem suggests the question was misread, misstated or overlooked.

Acceptance. The site's curator, Thomas Bloom, marks the problem SOLVED, records in the problem's commentary that GPT-5.4 Pro, prompted by Chojecki, proved that there is no uniform bound, and wrote on the thread (post 7373, 2026-07-06) that the result is an immediate corollary of the Erdős--Rado theorem, restating the construction: that curator acceptance is the reviewed evidence. A thread reader reported (post 5674) a check that found no issues. There is no refereed publication. The system is named as the thread and the commentary name it, GPT-5.4 Pro. A later note by gavinsherry (GitHub gist https://gist.github.com/gavinsherry/ad9b6f85e2afc0a830d015a6b5d6d52a, created 2026-04-27, linked from thread post 5935 of the same day and prepared, as it says, with AI assistance) gives an independent exposition and check of the same construction and of the answer $2$ for countable families; its addendum is recorded on the problem page.

Formalization. A Lean 4 development announced on the thread (post 6491, 2026-05-17), linked above at its commit of that day, declares itself a formalization based on the note. It proves the countable-sequence reading in full, that two colors suffice for any countable sequence of infinite sets, and proves the arbitrary-family theorem from the Erdős--Rado partition relation $\kappa\to(\omega)^2_\mu$ taken as an explicit hypothesis, which it does not formalize; it was developed with the assistance of OpenAI Codex 5.5, as its author states. A thread reader reported a check of it (post 6495). The development was not built or audited here, so the page lists no formalized evidence.