A combination of variable-temperature neutron scattering, reverse Monte Carlo analysis and direct Monte Carlo simulation is used to characterise the emergence of magnetic order in the metal-organic framework (MOF) Tb(HCOO)3 over the temperature range 100 K to 1.6 K = TN. We show that the magnetic transition at TN involves one-dimensional ferromagnetic ordering to a partially-ordered state related to the triangular Ising antiferromagnet. In this phase, the direction of magnetisation of ferromagnetic chains tends to alternate between neighbouring chains but this alternation is frustrated and is not itself ordered. In neutron scattering measurements this partial order gives rise to Bragg-like peaks, which cannot be interpreted using conventional magnetic crystallography without resort to unphysical spin models. The existence of low-dimensional magnetic order in Tb(HCOO)3 is stabilised by the contrasting strength of inter-and intra-chain magnetic coupling, itself a consequence of the underlying MOF architecture. Our results demonstrate how MOFs may provide an attractive if as yet under-explored platform for the realisation and investigation of low-dimensional physics.Low-dimensional magnets have long provided an important playground for the discovery and exploitation of unconventional physics [1]-from the earliest studies of soliton excitations in CsNiF 3 [2-4] to contemporary research into quantum information transport in spin-chain compounds [5]. The sensitivity of low-dimensional spin systems to small perturbations results in a rich diversity of phase transitions and complex ordering phenomena. By way of example, the Ising spin-chain compound Ca 3 Co 2 O 6 exhibits a variety of equilibrium and non-equilibrium states [6][7][8], characterised by e.g. long-wavelength incommensurate spin density modulations and field-induced magnetisation plateaux reminiscent of Hofstadter fractalisation [9,10]. In the field of low-dimensional magnetism, arguably the strongest scientific interest from both experimental and theoretical perspectives has always been in the limit of strict 1D order [11][12][13][14][15][16]. Yet even in canonical systems such as Ca 3 Co 2 O 6 the divergence of correlation lengths along 1D spin chains is always accompanied by full 3D magnetic order [8,17]. So the realisation and experimental characterisation of genuine partially-ordered low-dimensional spin states remain an important challenge in the field.It was in this context that we chose to study magnetic order in terbium(III) formate, Tb(HCOO) 3 . In metal-organic frameworks (MOFs) such as Tb(HCOO) 3 , magneticallyactive transition-metal or rare-earth ions are connected via organic ligands to form extended 3D framework structures. Because organic ligands can support superexchange interactions that span a broad energy range, and because framework design allows controlled incorporation of low-dimensional structural motifs (e.g. chains, ladders, layers), MOFs are natural candidate hosts for low-dimensional magnetism [18]. Indeed the magnetic response of...