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Barth and Schneider prove that for any two countable dense subsets $A$ and $B$ of $\mathbb R$ there is a transcendental entire function $f$, real on the real line, such that for real $x$

$$f(x)\in B \iff x\in A.$$

Taking $A=B=\mathbb Q$ gives an entire function that sends exactly the rational real numbers to rational values. A transcendental entire function is not a polynomial, so in particular it is not linear, and Problem 226 is answered in the affirmative. The paper's title records that the function can be taken monotone on the real line. In a second paper the same authors extend the construction to countable dense subsets of $\mathbb C$ and their complements (J. London Math. Soc. (2) 4, no. 3 (April 1972), 482--488, doi), a strengthening outside the real-line question the problem asks. The paper is not held in the library and its proof is not compiled in this corpus.

Formalization. The statement file of the formal-conjectures project (FormalConjectures/ErdosProblems/226.lean) marks the problem solved and points, through its formal_proof attribute, at a Lean 4 file in Boris Alexeev's lean-proofs repository, linked above at a pinned commit. That file declares itself a formalization of a solution to Problem 226 and names K. F. Barth, W. J. Schneider and ChatGPT as its informal authors and Aristotle and Boris Alexeev as its formal authors, so it is recorded on this page as a formalization of this claim rather than as an independent result. Its theorem erdos_226 asserts an entire function $F$, real on the real line, whose restriction to $\mathbb R$ is not affine and preserves rationality in both directions; the file contains no sorry and no axiom command at the pinned commit. This project has not built the file or audited its statement against the problem, so it supplies no formalized evidence; the site's label PROVED (LEAN) refers to this development.

Acceptance. The paper is refereed: K. F. Barth and W. J. Schneider, Entire functions mapping countable dense subsets of the reals onto each other monotonically, J. London Math. Soc. (2) 2, part 4 (October 1970), 620--626. The site's curator, Thomas F. Bloom, marks Problem 226 proved and credits this paper with the solution. The page is dated by the first day of the issue month, since the paper's first posting carries no finer date.