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Claim. The first question of Problem 1174 asks for a graph $G$ with no $K_4$ such that every coloring of its edges with countably many colors has a monochromatic triangle, in arrow notation $G\to(K_3)^2_{\aleph_0}$ with $K_4\not\le G$. Section 5 of Shelah's chapter addresses this question, which it calls an old one of Erdős and Hajnal, and proves the consistency of a slightly stronger statement. Lemma 5.1: if $\mu<\lambda<\kappa$, $\kappa$ is a measurable cardinal (or one of two weaker hypotheses the lemma states holds, one on $\kappa$ and one on $\lambda$, in the chapter's notation), $2\le m<\omega_1$ and $\lambda=\lambda^{<\lambda}$, then some $\lambda^+$-c.c., $\lambda$-complete forcing notion of power $\kappa$ forces $2^\lambda=\kappa$ and adds a graph $G$ of power $\kappa$ with $G\to(K_{k(*)})^2_\mu$ that embeds no $K_{k(*)+1}$. With $k(*)=3$ and $\mu=\aleph_0$ the extension has a $K_4$-free graph every countable edge coloring of which has a monochromatic triangle. So, relative to the consistency of ZFC with the lemma's hypothesis, ZFC does not refute the existence of such a graph: the first question is not disprovable in the site's sense. The section says that more on forbidden infinite subgraphs would appear later; the second question, with $K_{\aleph_1}$ forbidden and a monochromatic $K_{\aleph_0}$ demanded, is settled consistently by Komjáth and Shelah's edge partition theorem.

Covers. The first question of Problem 1174 (the part k4_free_graph), as a consistency statement: in a forcing extension built from a measurable cardinal, or from the lemma's weaker hypothesis, there is a $K_4$-free graph with $G\to(K_3)^2_{\aleph_0}$. It settles one side of that part only: ZFC does not refute the existence of such a graph, but whether ZFC can prove it is not settled, and one side alone leaves the question open. It does not cover the second question, nor the question whether such a graph exists in ZFC, which Komjáth's 2025 survey (Problem 53) records as open; in the commentary to his Problems 51 and 52 Komjáth notes that no graph of cardinality at most $\mathfrak c$ can have their properties, since $K_{\mathfrak c}$ is a union of $\aleph_0$ bipartite graphs, and the same bipartite argument shows that a $K_4$-free graph forcing a monochromatic triangle has more than $2^{\aleph_0}$ vertices, as the problem page's Known Results record from the binary-digit coloring. Whether the consistency is relative to ZFC alone is not recorded here: the lemma is stated from a measurable cardinal above $\lambda$ or from its two alternatives, and the remark of Komjáth and Shelah (1993) that measurables can be eliminated through §§3--4 of this chapter concerns their own theorem.

Source. Saharon Shelah, Consistency of positive partition theorems for graphs and models, in Set theory and its applications (Toronto, ON, 1987), J. Steprāns and S. Watson, eds., Lecture Notes in Mathematics 1401, Springer, Berlin, 1989, pp. 167--193; DOI 10.1007/BFb0097339; Shelah archive Sh:289, whose copy the archive labels the published version. The chapter's card is the source card, which also records that Problem 595 asks the first question in other words. This page rests on the introduction of §5 and the statement of Lemma 5.1, in the archive copy; the proof was not followed. The volume carries only the year, so this page is dated the first of January 1989.

Acceptance. Reviewed: the curator of erdosproblems.com, T. F. Bloom, labels the problem not disprovable, and the page's remark credits Shelah with the consistency of a graph with either property; the booklet [Va99, 7.91] that is the problem's source carries the same remark. The label was asked for in a thread comment of 19 March 2026, and the community database records it from that day; an earlier thread comment (9 February 2026) links the archive copies of the Komjáth--Shelah paper and of a Shelah survey. The chapter appeared in a Springer Lecture Notes in Mathematics proceedings volume; no evidence that the volume's chapters were refereed was found, so refereed is not listed. Nothing on this page is independently reviewed by this project.

Depends on. No other wiki page; the claim rests on the chapter above.