...To be serious, though, you would want some kind of steel. I am not going to say what kind of steel, though, as they all have their possibilities. The thing about steel that makes it so great is it's hardenability and the wide variety of crystal structures that can be created within the steel itself to give it different properties.
Martensite is what generally makes up a hamon on a Japanese sword.
Pearlite is soft and allows for give without being brittle.
Lower Bainite is springy, allowing a blade to bend and return with less permanent deformation as pearlite.
Upper Bainite is springy, but is considerably more rigid (therefore fragile) than lower bainite.
...There are tons more, (ferrite, cementite, austentite, etc) but either I don't know much about them or they don't appear in the finished product anyways.
These possibilities make steel the top choice for swords. You can selectively harden a sword to have a martensitic edge, being extremely hard, and a pearlitic core/spine to give it some springyness. You can harden the blade to a lower bainitic state to allow for really good durability.
All this, however, stems from the heat treatment of steel. To reach martensite, the steel has to be heated, then quenched to cool rapidly. Martensite is extremely hard, but also brittle. This means that the steel then has to be tempered to relieve the stresses. To relieve said stresses, you must soften the martensite slightly. Lower edge retention, but less brittleness. Add in, say, a bainitic core/spine to the equation and you have a blade that is springy, therefore durable, with a hard cutting edge, meaning good edge retention and "bite".
All of these structures can be coaxed from many of the different kinds of steel, so it is not about the type of steel as it is what the swordsmith does with the steel.
As far as immunity to breaking and edge retention, there has to be a balance of both, meaning there is a slight lack in both directions. If you want resistance to dulling, you need a really hard blade...the downside being that the harder steel is, the more brittle it is. If you want resistance to breaking, you need soft, springy steel...which means that you have lower edge retention.
There are pros and cons to everything. Equally important is that steel is and always will be subject to chemistry and the laws of physics.
Don't worry about finding the ultimate supersteel, because it won't happen. Look, instead, for the best swordsmith available who can coax the best out of whatever steel will be used for what kind of sword and what sort of targets it will be cutting.
(PS: Avoid titanium as both a base metal and an additive. Search titanium nitride embrittlement.)
Swordsman trained in Kashima Shinto Ryu and Mugai Ryu iai, sayashi (fittings maker), togishi (sword polisher), tsukamakishi (sword handle wrapper) and sword designer/consultant