We report the crystal structures and magnetic properties of two psuedo-polymorphs of the $S=1/2$ Ti$^{3+}$ coordination framework, KTi(C$_2$O$_4$)$_2cdot$xH$_2$O. Single-crystal X-ray and powder neutron diffraction measurements on $alpha$-KTi(C$_2$O$_4$)$_2cdot$xH$_2$O confirm its structure in the tetragonal $I4/mcm$ space group with a square planar arrangement of Ti$^{3+}$ ions. Magnetometry and specific heat measurements reveal weak antiferromagnetic interactions, with $J_1approx7$ K and $J_2/J_1=0.11$ indicating a slight frustration of nearest- and next-nearest-neighbor interactions. Below $1.8$ K, $alpha$ undergoes a transition to G-type antiferromagnetic order with magnetic moments aligned along the $c$ axis of the tetragonal structure. The estimated ordered moment of Ti$^{3+}$ in $alpha$ is suppressed from its spin-only value to $0.62(3)~mu_B$, thus verifying the two-dimensional nature of the magnetic interactions within the system. $beta$-KTi(C$_2$O$_4$)$_2cdot$2H$_2$O, on the other hand, realises a three-dimensional diamond-like magnetic network of Ti$^{3+}$ moments within a hexagonal $P6_222$ structure. An antiferromagnetic exchange coupling of $Japprox54$ K -- an order of magnitude larger than in $alpha$ -- is extracted from magnetometry and specific heat data. $beta$ undergoes Neel ordering at $T_N=28$ K, with the magnetic moments aligned within the $ab$ plane and a slightly reduced ordered moment of $0.79~mu_B$ per Ti$^{3+}$. Through density-functional theory calculations, we address the origin of the large difference in the exchange parameters between the $alpha$ and $beta$ psuedo-polymorphs. Given their observed magnetic behaviors, we propose $alpha$-KTi(C$_2$O$_4$)$_2cdot$xH$_2$O and $beta$-KTi(C$_2$O$_4$)$_2cdot$2H$_2$O as close to ideal model $S=1/2$ Heisenberg square and diamond lattice antiferromagnets, respectively.
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