Why not have everyone hold a copy of some arbitrarily old section of the history of posts (in their client, as a separate program on their device, whatever) with a choice to hold the entire history.
Then, you connect to some peers, add a post by communicating it with them, and they communicate with others and so on, adding it to their local copy, until everyone has the message? Essentially, everyone is a relay.
You could broadcast to some sunset of your peers and then wait for another subset to show the post to ensure that your message proliferated the network.
Obviously, this is extremely similar to how it works with cryptocurrencies, but the fact that there isn't a requirement that everyone must agree on what the chain looks like should make this process much simpler. You can just sort posts by timestamp and if some people don't have it, then so what. That's the trade-off for true decentralization.
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