The "Livingstone Formulation" (coined by David Hirsh) is the claim -- made in response to allegations of antisemitism -- that such allegations are made in bad faith as a means to silence or squelch all criticism of Israel.
It is an interesting fact about the Livingstone Formulation that the event which inspired it actually had nothing to do with Israel. Former London Mayor Ken Livingstone was accused of antisemitism after he compared a Jewish reporter to a "Nazi war criminal" and a "concentration camp guard". One would think the antisemitic nature of such statements could easily be disassociated from any particular beliefs about Israel, as the controversy had nothing to do with Israel at all. But Livingstone defended himself by declaring that "For far too long the accusation of antisemitism has been used against anyone who is critical of the policies of the Israeli government, as I have been".
In short, Livingstone took a non-Israel related instance of antisemitism and transformed it into a case of "criticism of Israel", then used it to complain about Jews who just couldn't tolerate criticism of Israel. From the get-go, the applicability of the "Livingstone Formulation" did not depend on the antisemitism in question actually being Israel-related.
There are many instances of this. Some are public: The courts which held that firebombing a synagogue was not antisemitic but criticism of Israel, and the guy on my twitter feed who fretted that a contrary decision "Sounds too much like: you can't criticise Israel because it's anti-Semitic." Some are private: The time I sent an anti-discrimination paper out for comments and one reader suggested removing the "Israeli foreign policy examples" (there were none, but there were antisemitism examples -- how was it that they got confused?).
Now consider this flyer, about prominent Labour activist and top Jeremy Corbyn ally Jackie Walker.
"To oppose Israel is not to be anti-semitic" (presumably that's Walker). Chomsky offers his own blurb of endorsement: "I wholeheartedly support the right of anyone to criticise Israel without being branded anti-semitic."
All of this might make one wonder what it was that Jackie Walker did that brought these terrible, horrible, no-good, clearly unfair accusations of antisemitism?
Well, she claimed that Jews were "the chief financiers of the slave trade." Then she criticized Holocaust Memorial Day for being exclusionary to other victims of mass atrocity (#AllGenocidesMatter). Finally, she questioned the need for security at Jewish schools and institutions, suggesting Jewish concerns about being targeted were exaggerated or embellished.
None of these are Israel-related. Yet Walker recasts the debate as one about the freedom to criticize Israel, and her backers enthusiastically endorse the transformation. Clearly, they have a point: if one can't contend that Jews were chief financiers of the slave trade hundreds of years before Israel was established, what possible space is there to criticize Israel?
There is an element of farce to this. It does not squelch pro-Palestinian advocacy to call efforts to tie Jews to the slave trade antisemitic. Such superficially ludicrous arguments only work because, for all the claims about people who conflate "criticism of Israel" with "antisemitism", there seem to be far more who conflate "criticism of antisemitism" with "intolerance of criticism of Israel". Were it not for that belief -- the cleansing power of anti-Zionism -- we would not see people try to take things that are transparently not about Israel and convert them into it. If everything is "criticism of Israel", then nothing can be antisemitic -- because what is "antisemitism" but the bad faith effort to silence critics of Israel?