Is there any risk of spamming the mempool by continuously updating the transaction fee?
Nodes should only be propagating txs increasing the fee, but yeah, the possibility of spamming the network through increasing the fee by 1 sat steps seems to be worse.. But users can already do it with RBF transactions. Only ancient pre-RBF nodes would not propagate the tx.
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Nodes do not propagate RBF replacements unless they pay a minimum threshold of additional funds per byte. The cost is high enough that it'd be a lot cheaper to just buy cloud servers and DoS attack the entire P2P network directly.
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