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The phrase "a common message running" is correct and usable in written English.
It can be used to describe a recurring theme or idea that is present throughout a text or conversation.
Example: "In the novel, there is a common message running through the characters' journeys about the importance of friendship and loyalty."
Alternatives: "a recurring theme" or "a shared message".
Exact(1)
Their manner is more elegantly elaborate than the name suggests, but there is a common message running plainly through their core, the persuasion of the young man addressed, Mr WH (William Herbert, Henry Wriothesley, William Holme, or none of the above), to settle down with a wife and make babies.
Similar(59)
We recall that in this work, message w2 is a common message to both receivers, and w1 is a private message to user 1.
But honing a common message for all 222 House Republicans has been an elusive goal.
That is not a common message to hear from an American pulpit these days.
Ultimately, your stories should share a common message — it's okay to step up and out.
We consider the most general scenario where each user receives both an independent message and a common message.
The transmitter wishes to send a common message to two receivers and a confidential message to receiver 1.
This model has been studied in [17 19] for the case of two confidential messages and in [20] for the case of two confidential messages and a common message.
Underneath the "humor," she noted, is a common message.
Underneath the "humor," she noted, is a common message. .
A common message board to (and from) which messages are posted (and read) assists the agents in communicating with each other.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com