Claim: every Jordan curve in \(\mathbb{R}^2\\) contains four points that are the vertices of a (possibly tilted, possibly tiny) square. Proved for piecewise-analytic, symmetric and many smooth cases; Greene–Lobb (2020) settled all smooth Jordan curves. The remaining open case is curves of low regularity — merely continuous.
Toeplitz asked in 1911. True for piecewise-analytic curves (Emch), Lipschitz and star-shaped curves, curves of bounded variation… yet no proof covers even all smooth curves. Schnirelmann's 1929 claimed proof had a gap repaired only for a subclass. The square peg resists precisely because it demands finite combinatorial structure from arbitrary topology.
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