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Dear Teachers,

1. - Can we at least open the window and get some ventilation here?

- Knock yourself out

- Thanks. I will.

- What does "knock yourself out" mean here?

2. - I've got to finish a report by ten, and then I'll have to attend a senior staff meeting. Then I'll have a meeting with my colleagues over lunch.

- Wow! That's cutting it close.

- What does "that's cutting it close" mean here?

Thanks a million to Teachers,

Stevenukd.
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By combining what the others have said, in perhaps a more explicit sentence:

Sure, go ahead, knock yourself out, and open it (the window).
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Yes, "knock yourself out" is another way of saying "whatever makes you happy." It's extremely casual.
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Comments  
1. - Can we at least open the window and get some ventilation here?

- Knock yourself out

- Thanks. I will.

- What does "knock yourself out" mean here? -- Exert your maximum effort; try very hard.

2. - I've got to finish a report by ten, and then I'll have to attend a senior staff meeting. Then I'll have a meeting with my colleagues over lunch.

- Wow! That's cutting it close.

- What does "that's cutting it close" mean here? -- That is very tight scheduling.
For number one, it was more of a sarcastic statement, which would translate to "Sure, go ahead."
 Marius Hancu's reply was promoted to an answer.
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 BarbaraPA's reply was promoted to an answer.