Counting Families of Mutually Intersecting Sets
نویسندگان
چکیده
We determine the number of maximal intersecting families on a 9-set and find 423295099074735261880. We determine the number of independent sets of the Kneser graph K(9, 4) and find 366996244568643864340. Finally, we determine the number of intersecting families on an 8-set and find 14704022144627161780744368338695925293142507520.
منابع مشابه
Maximal Intersecting Families of Finite Sets and «uniform Hjelmslev Planes
The following theorem is proved. The collection of lines of an n-uniform projective Hjelmslev plane is maximal when considered as a collectiion of mutually intersecting sets of equal cardinality.
متن کاملCounting Intersecting and Pairs of Cross-Intersecting Families
A family of subsets of {1, . . . , n} is called intersecting if any two of its sets intersect. A classical result in extremal combinatorics due to Erdős, Ko, and Rado determines the maximum size of an intersecting family of k-subsets of {1, . . . , n}. In this paper we study the following problem: how many intersecting families of k-subsets of {1, . . . , n} are there? Improving a result of Bal...
متن کاملEnumeration of intersecting families
For a natural number m, let m =11,2, . . ., m} . An intersecting family on m is a set A of sets such that U A g m and any two members of A have non-empty intersection . We let J_ be the set of all maximal intersecting families on m . We are concerned with estimating JJ . J . In Section 2 we obtain a lower bound by elementary counting methods . In Section 3 we obtain an upper bound using a resul...
متن کاملRegular bipartite graphs and intersecting families
In this paper we present a simple unifying approach to prove several statements about intersecting and cross-intersecting families, including the Erdős–Ko–Rado theorem, the Hilton–Milner theorem, a theorem due to Frankl concerning the size of intersecting families with bounded maximal degree, and versions of results on the sum of sizes of non-empty cross-intersecting families due to Frankl and ...
متن کاملStructure and properties of large intersecting families
We say that a family of k-subsets of an n-element set is intersecting, if any two of its sets intersect. In this paper we study different extremal properties of intersecting families, as well as the structure of large intersecting families. We also give some results on k-uniform families without s pairwise disjoint sets, related to Erdős Matching Conjecture. We prove a conclusive version of Fra...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Electr. J. Comb.
دوره 20 شماره
صفحات -
تاریخ انتشار 2013