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Ahlswede–Khachatrian Theorem
In extremal set theory, the Ahlswede–Khachatrian theorem generalizes the Erdős–Ko–Rado theorem to -intersecting families. Given parameters , and , it describes the maximum size of a -intersecting family of subsets of \ of size , as well as the families achieving the maximum size. Statement Let n \ge k \ge t \ge 1 be integer parameters. A ''-intersecting family'' is a collection of subsets of \ of size such that if then , A\cap B, \ge t. Frankl Frankl is a surname. Notable people with the surname include: * Ludwig August von Frankl (1810–1894), Austrian writer and philanthropist * Michal Frankl (born 1974), Czech historian * Nicholas Frankl (born 1971), British-Hungarian entrepreneur ... constructed the -intersecting families : \mathcal_ = \. The Ahlswede–Khachatrian theorem states that if is -intersecting then : , \cal F, \leq \max_ , \mathcal_, . Furthermore, equality is possible only if is ''equivalent'' to a Frankl family, meaning that it coincide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extremal Set Theory
Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vectors, sets, etc.) can be, if it has to satisfy certain restrictions. Much of extremal combinatorics concerns classes of sets; this is called extremal set theory. For instance, in an ''n''-element set, what is the largest number of ''k''-element subsets that can pairwise intersect one another? What is the largest number of subsets of which none contains any other? The latter question is answered by Sperner's theorem, which gave rise to much of extremal set theory. Another kind of example: How many people can be invited to a party where among each three people there are two who know each other and two who don't know each other? Ramsey theory shows that at most five persons can attend such a party (see Theorem on Friends and Strangers). Or, suppose we are given a finite set of nonzero int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erdős–Ko–Rado Theorem
In mathematics, the Erdős–Ko–Rado theorem limits the number of Set (mathematics), sets in a family of sets for which every two sets have at least one element in common. Paul Erdős, Chao Ko, and Richard Rado proved the theorem in 1938, but did not publish it until 1961. It is part of the field of combinatorics, and one of the central results of The theorem applies to families of sets that all have the same and are all subsets of some larger set of size One way to construct a family of sets with these parameters, each two sharing an element, is to choose a single element to belong to all the subsets, and then form all of the subsets that contain the chosen element. The Erdős–Ko–Rado theorem states that when n is large enough for the problem to be nontrivial this construction produces the largest possible intersecting families. When n=2r there are other equally-large families, but for larger values of n only the families constructed in this way can be largest. The Erd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Péter Frankl
Péter Frankl (born 26 March 1953 in Kaposvár, Somogy County, Hungary) is a mathematician, street performer, columnist and educator, active in Japan. Frankl studied mathematics at Eötvös Loránd University in Budapest and submitted his PhD thesis while still an undergraduate. He holds a PhD degree from the University Paris Diderot as well. He has lived in Japan since 1988, where he is a well-known personality and often appears in the media. He keeps travelling around Japan performing (juggling and giving public lectures on various topics). Frankl won a gold medal at the International Mathematical Olympiad in 1971. He has seven joint papers with Paul Erdős, and eleven joint papers with Ronald Graham. His research is in combinatorics, especially in extremal combinatorics. He is the author of the union-closed sets conjecture. Personality Both of his parents were survivors of concentration camps and taught him "The only things you own are in your heart and brain". So he beca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyula O
Gyula may refer to: * Gyula (title), Hungarian leader title in the 9th–10th centuries * Gyula (name), Hungarian male given name, derived from the title ; People * Gyula II, the Hungarian ''gyula'' who ruled Transylvania in the 10th-century and was baptized in Constantinople around 950 * Gyula III, the ''gyula'' who ruled Transylvania and was defeated by his maternal uncle, King Stephen I of Hungary around 1003 ; Places * Gyula, Hungary, town in Hungary * Gyulaháza, village in Hungary * Gyulakeszi, village in Hungary * , Hungarian name of Alba Iulia, city in Romania, the former seat of the Transylvanian ''gyulas'' {{disambiguation, hn, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irit Dinur
Irit Dinur () is an Israeli computer scientist. She is professor of computer science at the Weizmann Institute of Science. In 2024 she was appointed a permanent faculty member in the School of Mathematics of the Institute for Advanced Study. Her research is in foundations of computer science and in combinatorics, and especially in probabilistically checkable proofs and hardness of approximation. Biography Irit Dinur earned her doctorate in 2002 from the school of computer science in Tel Aviv University, advised by Shmuel Safra; her thesis was entitled ''On the Hardness of Approximating the Minimum Vertex Cover and The Closest Vector in a Lattice''. She joined the Weizmann Institute after visiting the Institute for Advanced Study in Princeton, New Jersey, NEC, and the University of California, Berkeley. Dinur published in 2006 a new proof of the PCP theorem that was significantly simpler than previous proofs of the same result. Awards and recognition In 2007, she was given the M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shmuel Safra
Shmuel (Muli) Safra () is an Israeli computer scientist. He is a professor of computer science at Tel Aviv University, Israel. He was born in Jerusalem. Safra's research areas include complexity theory and automata theory. His work in complexity theory includes the classification of approximation problems—showing them NP-hard even for weak factors of approximation—and the theory of probabilistically checkable proofs (PCP) and the PCP theorem, which gives stronger characterizations of the class NP, via a membership proof that can be verified reading only a constant number of its bits. His work on automata theory investigates determinization and complementation of finite automata over infinite strings, in particular, the complexity of such translation for Büchi automata, Streett automata and Rabin automata. In 2001, Safra won the Gödel Prize in theoretical computer science for his papers "Interactive Proofs and the Hardness of Approximating Cliques" and "Probabilistic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Families Of Sets
Family (from ) is a group of people related either by consanguinity (by recognized birth) or affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictability, structure, and safety as members mature and learn to participate in the community. Historically, most human societies use family as the primary purpose of attachment, nurturance, and socialization. Anthropologists classify most family organizations as matrifocal (a mother and her children), patrifocal (a father and his children), conjugal (a married couple with children, also called the nuclear family), avuncular (a man, his sister, and her children), or extended (in addition to parents, spouse and children, may include grandparents, aunts, uncles, or cousins). The field of genealogy aims to trace family lineages through history. The family is also an important economic unit studied in family economics. The word "families" can be used metaphorically t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at which they meet. More generally, in set theory, the intersection of sets is defined to be the set of elements which belong to all of them. Unlike the Euclidean definition, this does not presume that the objects under consideration lie in a common space. It simply means the overlapping area of two or more objects or geometries. Intersection is one of the basic concepts of geometry. An intersection can have various geometric shapes, but a point is the most common in a plane geometry. Incidence geometry defines an intersection (usually, of flats) as an object of lower dimension that is incident to each of the original objects. In this approach an intersection can be sometimes undefined, such as for paral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Discrete Mathematics
In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasonin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial And Binomial Topics
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book ''Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential function an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
