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Ramsey's Theorem
By this time, you are probably not surprised to see that there is a very general form of Ramsey's theorem. We have a bounded number of bins or colors and we are placing the subsets of a fixed size into these categories.
Ramsey's Theorem
By this time, you are probably not surprised to see that there is a very general form of Ramsey's theorem. We have a bounded number of bins or colors and we are placing the subsets of a fixed size into these categories. The conclusion is that there is a large set which is treated uniformly.
Here's the formal statement.
We don't include the proof of this general statement here, but the more ambitious students may attempt it on their own. Note that the case \(s=1\) is just the , while the case \(s=r=2\) is just . An argument using double induction is required for the proof in the general case. The first induction is on \(r\) and the second is on \(s\).
Zvino iwe Hapana calculator inogadzirisa iyi imwe, asi mapepa ayo anogona kuongororwa. Tsananguro:
Mubhadharo wakasununguka unowedzera mazita ezvidzidzo, zvinyorwa zvezvidzidzo zvawakati wazvigadzirira, zvidzidzo zvawave uchizvigadzirisa, uye mudzidzisi waungabvunza nezvechidzidzo ichi. Maths inowanikwa kune vese, vasiri vadzidzi kana vadzidzi. Kuwana mazita ezvidzidzo zvinodiwa kunogona kuitwa pawebsite ye Maths.
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Mabvunzo vanhu vanobvunza
What is the difference between probability and statistics?
Probability goes from a known model to what the data should look like; statistics goes from data back to the model. Probability theory is the deductive half.
What does the law of large numbers promise?
That the average of many independent samples converges to the expected value. It says nothing about any single trial.
Zvimwe zvenyaya iyi zvakashandurwa kubva Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensed uye re-akaratidza pano; zvikanganiso ndedu.
More in Probability Theory
Sample spaces and the axiomsRandom variables and expectationThe common distributionsThe law of large numbers and the central limit theorem