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Here we present the structural and magnetic properties of the new honeycomb material ${\mathrm{Ag}}_{3}{\mathrm{LiMn}}_{2}{\mathrm{O}}_{6}$. The system $\mathrm{Ag}[{\mathrm{Li}}_{1/3}{\mathrm{Mn}}_{2/3}]{\mathrm{O}}_{2}$ belongs to a quaternary $3R$-delafossite family and crystallizes in a monoclinic symmetry with space group $C\phantom{\rule{0.16em}{0ex}}2/m$, and the magnetic ${\mathrm{Mn}}^{4+}\phantom{\rule{0.28em}{0ex}}(S=3/2)$ ions form a honeycomb network in the $ab$ plane. An anomaly around 50 K and the presence of antiferromagnetic (AFM) coupling (Curie-Weiss temperature ${\ensuremath{\theta}}_{CW}\ensuremath{\sim}\ensuremath{-}51\phantom{\rule{0.28em}{0ex}}\mathrm{K}$) were inferred from our magnetic susceptibility data. The magnetic specific heat clearly manifests the onset of magnetic ordering in the vicinity of 48 K, and the recovered magnetic entropy, above the ordering temperature, falls short of the expected value, implying the presence of short-range magnetic correlations. An asymmetric Bragg peak (characteristic of two-dimensional order), seen in neutron diffraction, gains intensity even above the ordering temperature, thus showing the existence of short-range spin correlations. Our electron spin resonance (ESR) experiments corroborate the bulk magnetic data. Additionally, the ESR line broadening upon approaching the ordering temperature ${T}_{\mathrm{N}}$ could be described in terms of a Berezinskii-Kosterlitz-Thouless scenario with ${T}_{\mathrm{KT}}=40(1)\phantom{\rule{0.28em}{0ex}}\mathrm{K}$. The $^{7}\mathrm{Li}$ NMR line shift probed as a function of temperature tracks the static susceptibility (${K}_{\mathrm{iso}}$) of magnetically coupled ${\mathrm{Mn}}^{4+}$ ions. The $^{7}\mathrm{Li}$ spin-lattice relaxation rate ($1/{T}_{1}$) exhibits a sharp decrease below about 50 K. A critical divergence is absent at the ordering temperature perhaps because of the filtering out of the antiferromagnetic fluctuations at the Li site, i.e., at the centers of the hexagons in the honeycomb network. Combining our bulk and local probe measurements, we establish the presence of an ordered ground state for the honeycomb system ${\mathrm{Ag}}_{3}{\mathrm{LiMn}}_{2}{\mathrm{O}}_{6}$. Our ab initio electronic structure calculations suggest that in the $ab$ plane, the nearest-neighbor (NN) exchange interaction is strong and AFM, while the next-NN and the third-NN exchange interactions are FM and AFM, respectively. The interplanar exchange interaction is found to be relatively small. In the absence of any frustration the system is expected to exhibit long-range, AFM order, in agreement with experiment.

We have investigated the thermodynamic and local magnetic properties of the Mott insulating system Ag$_{3}$LiRu$_{2}$O$_{6}$ containing Ru$^{4+}$ (4$d$$^{4}$) for novel magnetism. The material crystallizes in a monoclinic $C2/m$ structure with RuO$_{6}$ octahedra forming an edge-shared two-dimensional honeycomb lattice with limited stacking order along the $c$-direction. The large negative Curie-Weiss temperature ($\theta_{CW}$ = -57 K) suggests antiferromagnetic interactions among Ru$^{4+}$ ions though magnetic susceptibility and heat capacity show no indication of magnetic long-range order down to 1.8 K and 0.4 K, respectively. $^{7}$Li nuclear magnetic resonance (NMR) shift follows the bulk susceptibility between 120-300 K and levels off below 120 K. Together with a power-law behavior in the temperature dependent spin-lattice relaxation rate between 0.2 and 2 K, it suggest dynamic spin correlations with gapless excitations. Electronic structure calculations suggest an $S = 1$ description of the Ru-moments and the possible importance of further neighbour interactions as also bi-quadratic and ring-exchange terms in determining the magnetic properties. Analysis of our $\mu$SR data indicates spin freezing below 5 K but the spins remain on the borderline between static and dynamic magnetism even at 20 mK.