As in, do relay nodes need to keep a record of the transactions they've
relayed? Strictly speaking, the answer is no: one a tx is relayed modulo
DoS concerns the entire thing can be discarded by the node. (unconfirmed
txs spending other unconfirmed txs can be handled by creating packages
of transactions, evaluated as a whole)

To mitigate DoS concerns, we of course have to have some per-UTXO limit
on bandwidth relayed, but that task can be accomplished by simply
maintaining some kind of per-UTXO record of bandwidth used. For instance
if the weighted fee and fee/KB were recorded, and forced to - say -
double for each additional tx relayed that spent a given UTXO you would
have a clear and simple upper limit of lifetime bandwidth. Equally it's
easy to limit bandwidth moment to moment by asking peers for highest
fee/KB transactions they advertise first, stopping when our bandwidth
limit is reached.

You probably could even remove IsStandard() pretty much entirely with
the right increasingly expensive "replacement" policy, relying on it
alone to provide anti-DoS. Obviously this would simplify some of the
debates around mining policy! This could even be re-used for scalable a
general-purpose messaging network paid by coin ownership if the UTXO set
is split up, and some kind of expiration over time policy is
implemented.

Miners of course would still want to have a mempool, but that codebase
may prove simpler if it doesn't have to work double-duty for relaying as
well.