How can I detect and warn that a certain user is sending many messages in a row?

I suggest saving the session the date the message was sent and a message counter. If the user sends another message, compare the date with which it is stored in the session, if the time difference is short, increment a counter in the session.

If this counter reaches a limit (eg 3 messages) refuse the next requests until a certain time has passed.

To save the time of the last post

$_SESSION["LastPost"] = time();

To compare the time of the post

// se o post foi feito novamente nos últimos 10 segundos
if (isset($_SESSION["LastPost"])
    && $_SESSION["LastPost"] <= (time() - 10))
{
    if (!isset($_SESSION["PostCount"]))
        $_SESSION["PostCount"] = 0;

    $_SESSION["PostCount"] += 1;
}

And to check if the user has reached the posts per second limit

if (isset($_SESSION["PostCount"]) && $_SESSION["PostCount"] == 3)
{
    // você pode setar uma trava
    $_SESSION["LockPost"] = time() + 60;
}

If the lock exists ignore the request.

if (isset($_SESSION["LockPost"]) && $_SESSION["LockPost"] >= time())
{
    // bloqueado, ignore
}
else if (isset($_SESSION["LockPost"]))
{
    // o tempo de bloqueio passou
    // zere todas as variáveis da sessão
    unset($_SESSION["LockPost"]);
    // ...
}