Okay, let's try something.
Open up the wp-content/plugins/ajax-search-pro/ajax-search-pro.php file and go to line 13, where you should see this:
change that line to:
define("ASP_PATH", untrailingslashit(plugin_dir_path(__FILE__)) );
Now try to activate the plugin again, if it throws the same error message, then it's something else. Copy/pase the error message you get after this change.Best,
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