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High energy e$^+$e$^-$ colliders offer unique possibility for the most general dark matter search based on the mono-photon signature. Analysis of the energy spectrum and angular distributions of photons from the initial state radiation can be used to search for hard processes with invisible final state production. Most studies in the past focused on scenarios assuming heavy mediator exchange. We notice however, that scenarios with light mediator exchange are still not excluded by existing experimental data, if the mediator coupling to Standard Model particles is very small. We proposed a novel approach, where the experimental sensitivity to light mediator production is defined in terms of both the mediator mass and mediator width. This approach is more model independent than the approach assuming given mediator coupling values to SM and DM particles. Summarised in this contribution are published results of our studies concerning simulation of mono-photon events with WHIZARD and the expected sensitivity of the International Linear Collider (ILC) and Compact Linear Collider (CLIC) experiments to dark matter production.