How do we proceed after a 3-level direct raise? No game invitation after this start. The only auctions in which opener will require cooperation from responder as slam auctions. Let us see how to deal with them.
The auction 1M-3M is very well defined in strength and in the quality of the support. The exact range varies from partnership to partnership, and it depends on whether the pair uses Bergen or some other kind of conventional raise at the 3-level. But common features are that the trump suit is firmly established (being at least 5-4), and that 3NT is better used as artificial (for that exact reason).
In this scenario, a general principle kicks in. This principle is not very well described in the bridge literature (to my knowledge) but it is still quite important: when you are inviting a limited hand to slam, you should bid your shortness (0-1 cards in the suit). This principle works because the question being asked of a limited hand is not "are you interested in slam", it is rather "does your hand fit well with mine", and knowing the asker's shortness is the most direct bit of information to help responder to answer this question. Bridge is a game of compressed information -- you have limited space to transmit data and should, therefore, emphasize the information that allows partner to best evaluate the combined power of the two hands.
If you have raised to 3M with a typical hand such as Kxxx Qxxx xx Kxx and partner shows (say) a side club suit, that Queen of hearts may help him (picture AQxxxx Kx A AQxx) or not (switch the red suits). If he showed a side diamond suit, that King of clubs may be important or not. And so on. In other words, showing your long suit helps if you have help in that suit, but it doesn't help in the evaluation of the entire hand. Showing shortness does help you to evaluate honors in all side 3 suits.
Of course, sometimes a side Queen will be worth a lot, sometimes it will be worth little. We have limited space between 3M and 4M, so we have to follow the odds, and showing shortness helps with this wholesale evaluation. Responder will make a positive signal whenever the singleton helped his hand -- and this includes, if necessary, bypassing game.
So, in the auction 1S-3S-4D, responder should go beyond 4S if he likes the diamond shortness and lacks a heart control. (And if he has a heart control, he should be inclined to reopen the bidding over 4S by partner -- if his hand fits well with a diamond shortness opposite). In a sense, opener's bid of a new suit indicates that opener can visualize hands (with a good fit of side honors opposite his hand) which will make slam, even though responder is weak, and responder must trust that -- going to the five-level to ensure proper investigation when responder knows the singleton fits well is no great risk.
What do you do if you do not have a singleton, but still want to investigate slam? In this (rare) situation, you bid 3NT, which shows exactly this kind of hand (it is unlimited, therefore forcing, but does not depict the intention of playing in NT). A typical hand for this auction would be AKxxxx Ax Kxx AQ. Partner will bid any side control and you will both proceed having the knowledge of no singleton in opener's hand.
This is a good opportunity to dispel a misconception that I have witnessed among some players. The Serious/Non-Serious 3NT is a convention (which I will examine in a later article) designed to better limit the strength of yet-unlimited hands. It is never used if one hand already limited itself!
Note that to be limited is not the same as being weak. You can be limited and strong. Consider the auction 1m - 1M - 3M, for example -- opener limited his hand (3M is not forcing), but has still shown a good hand. And all the principles presented in this article are still operational. Any new suit (including 4m! To give a concrete auction: 1C - 1S - 3S - 4C) should be shortness, and 3NT should be slam interest without shortness.
Other raises at the 3-level
The same meanings should be used if the raise is limited. For example, after 1S - 3D (Bergen), you should still reserve 3NT to show a slam invitation without a singleton, and bid any singleton directly. If you have the ability to jump in a side suit (4H in this example), you can distinguish between a singleton and a void (which is immensely useful not only for hand evaluation but also for keycard asking).
Unlimited (or wide-ranging) raises to the 3-level (e.g. a Jacoby 2NT) should not follow this procedure. I have already written an article about my preferred major-suit raising structure. Check it out here.
When hearts are trumps
It is slightly more efficient to always use the next step (3S rather than 3NT) as the "no singleton" slam try, and to replace the meaning which would be ascribed to that bid (singleton spade) to 3NT. This replacement also should be made for responder's reaction: 1H - 3H - 3S - 3NT is a spade cuebid.
Other examples of auctions in which this principle should be operative
The basic example we have been looking at is 1S - 3S, but other examples include:
1C - 1H - 3H
1NT - 2H - 3S
2NT - 3H - 4S (if responder wishes to reopen here without asking for keycards, his new suits should show shortness)
And similar auctions in which the last person to bid has already limited his hand, and 3NT is not an option to play.