How can I set up a single wireless network through a number of WIFI routers?

Just set each router to use the same SSID but a different channel. Users will see it as a single wireless network when they move between floors. The different channels will ensure that the routers do not conflict with each other.

The wireless access points must be in bridge mode and dumping the wireless traffic into the same vlan (layer 2 broadcast domain).

They must have the same ssid. They should be on different channels. With 802.11b/g, even though there are a bunch of channels, you should stick to channels 1, 6, and 11 and not have adjacent APs have the same channels. http://en.wikipedia.org/wiki/IEEE_802.11#Channels_and_international_compatibility

If you leave the APs in router mode, each AP will be creating an isolated wireless network with it's own address space and when a client roams from one to another their arp tables and IP addresses may not be reasonable for the new network.

What you're talking about is a wireless routing mesh, which provides wireless coverage wherever you go for the same "wireless network". Depending on the capability of the device, this may or may not be possible - some of the cheaper routers will not accomodate the necessary changes needed to support a mesh.

I do know that there is purpose-built hardware for this. For instance, my work has a set of Motorola WS5100 units that provide coverage with 12 access points scattered throughout the building, but it requires specialized hardware.

If you are planning on doing this with commodity hardware, you will want something that supports:

• both ad-hoc and infrastructure mode (ad-hoc supports point-to-point mesh routing);
• you'll want to set all stations to the same SSID;
• use the same shared key;
• force the APs to function as strict routers, and not provide any other network services (can you imagine the chaos of having 10+ points providing their own DHCP grants?);
• you'll probably want something that has adjustable power output, to help reduce overlap;
• and probably, you'll want a 2nd authentication factor, such as RADIUS or certificates or something else, "just in case".

Of course, it wouldn't hurt to have some or all of those access points tied back to a switch somehow, but if you're doing a wireless mesh, this won't always be necessary - the mesh will "forward" the packets from stand-alone units as needed. And there lies the rub - getting the mesh to forward to the correct location.