The C10M Problem is about how on a modern server, you should be able to easily handle 10M concurrent connections with solid throughput and low jitter. Handling that level of traffic generally requires a more specialized approach than is offered by a stock Linux kernel.
Using a stock debian-8 image and a Go server you can handle 10M concurrent connections with low throughput and moderate jitter if the connections are mostly idle. The server design for this example is just about the simplest websocket server that is useful for anything. It is similar to a push notification server like the iOS Apple Push Notification Service, but without the ability to store messages if the client is offline.
The server accepts websocket connections ports 10000-11000 (to avoid exhaustion of ephemeral ports on the clients during testing) and in the url the client specifies a channel to connect to, such as: