As I see you don't really use a partial form. You might have been defined it as partial, but it is just a layout-less normal view. Since ajax form is doing a callback to an action, you should pass the message as result of the callback.
Read this one for a short but really intuitive description of Ajax.BeginForm's usage in comparison with Html.BeginForm:
Html.BeginForm() vs Ajax.BeginForm() in MVC3[
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