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Small Ramsey Numbers
Actually determining the Ramsey numbersR(m,n)Ramsey number R(m,n) referenced in seems to be a notoriously difficult problem, and only a handful of these values are known precisely.
Small Ramsey Numbers
Actually determining the Ramsey numbers\(R(m,n)\)Ramsey number \(R(m,n)\) referenced in seems to be a notoriously difficult problem, and only a handful of these values are known precisely. In particular, \(R(3,3)=6\) and \(R(4,4)=18\), while \(43\le R(5,5)\le 49\). The distinguished Hungarian mathematician Paul Erdős said on many occasions that it might be possible to determine \(R(5,5)\) exactly, if all the world's mathematical talent were to be focused on the problem. But he also said that finding the exact value of \(R(6,6)\) might be beyond our collective abilities.
In the following table, we provide information about the Ramsey numbers \(R(m,n)\) when \(m\) and \(n\) are at least \(3\) and at most \(9\). When a cell contains a single number, that is the precise answer. When there are two numbers, they represent lower and upper bounds.
For additional (or more current) data, see Dynamic Survey #DS1: Small Ramsey Numbers by Stanisław Radziszowski in the Electronic Journal of Combinatorics. ( was last updated using the 12 January 2014 version of that article.)
Энди сизлар Буни ҳисоблаш учун ҳисоблагич керак эмас, лекин унинг қисмлари ҳисоблаш мумкин. Қуйида бирор нарсани синаб кўринг ёки ўзингиз ёзинг.
Бепул ҳисоб ҳар бир дарсга ёзув қўшиб беради, сиз тугатган нарсалар ва ҳал қилинган масалалар бир жойда қайд этилади ва сиз бу саҳифа ҳақида устоздан сўрашингиз мумкин. Математика ўзида очиқ бўлиб, унга кириш ёки киришни истамаган ҳар кимга очиқ.
Ройхатдан ўтиш КиришБу ерда қўлланилган белгилар
Барча белгилар учун тўлиқ таъриф, сурат ва ҳар бир ҳарфнинг маъносини кўриш учун белгини босинг.
Одамлар сўрайдиган саволлар
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.
Бу саҳифанинг қисмлари Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). (Ушбу оятда Аллоҳ таоло Пайғамбаримиз Муҳаммадга (с. а. в.) Қуръони Каримни туширишни амр қилмоқда.)
Кўпроқ Probability Theory
Sample spaces and the axiomsRandom variables and expectationThe common distributionsThe law of large numbers and the central limit theorem