Unavoidable subhypergraphs: a-clusters

One of the central problems of extremal hypergraph theory is the description of unavoidable subhypergraphs, in other words, the Turan problem. Let a=(a"1,...,a"p) be a sequence of positive integers, k=a"1+...+a"p. An a-partition of a k-set F is a partition in the form F=A"1@?...@?A"p with |A"i|=a"i for 1=p and sufficiently large n, if F is a k-uniform family on n vertices with |F| exceeding the Erdos-Ko-Rado bound (n-1k-1), then F contains an a-cluster. The only extremal family consists of all the k-subsets containing a given element.

[1]  Dhruv Mubayi.  Erdos-Ko-Rado for three sets , 2006, J. Comb. Theory, Ser. A.

[2]  János Pach,et al.  On disjointly representable sets , 1984, Comb..

[3]  Vašek Chvátal.  An Extremal Set‐Intersection Theorem , 1974 .

[4]  Zoltán Füredi,et al.  Forbidding Just One Intersection , 1985, J. Comb. Theory, Ser. A.

[5]  Dhruv Mubayi,et al.  On the VC-dimension of uniform hypergraphs , 2007 .

[6]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[7]  Dhruv Mubayi,et al.  Set systems with union and intersection constraints , 2009, J. Comb. Theory, Ser. B.

[8]  Hanfried Lenz,et al.  Design theory , 1985 .

[9]  Zoltán Füredi,et al.  Exact solution of some Turán-type problems , 1987, J. Comb. Theory, Ser. A.

[10]  D. Mubayi.  An intersection theorem for four sets , 2007 .

[11]  Rudolf Ahlswede,et al.  Counterexample to the Frankl-Pach conjecture for uniform, dense families , 1997, Comb..

[12]  Peter Frankl,et al.  On Sperner Families Satisfying an Additional Condition , 1976, J. Comb. Theory, Ser. A.

[13]  Zoltán Füredi,et al.  On finite set-systems whose every intersection is a Kernel of a star , 1983, Discret. Math..

[14]  Jiuqiang Liu,et al.  Families of Sets with Intersecting Clusters , 2009, SIAM J. Discret. Math..

[15]  Zoltán Füredi,et al.  A new generalization of the Erdős-Ko-Rado theorem , 1983, Comb..

[16]  Vojtech Rödl,et al.  On a Packing and Covering Problem , 1985, Eur. J. Comb..

[17]  Dhruv Mubayi,et al.  Set systems without a simplex or a cluster , 2010, Comb..

[18]  P. Erdös,et al.  INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS , 1961 .

[19]  Tao Jiang,et al.  Set Systems without a Strong Simplex , 2010, SIAM J. Discret. Math..

[20]  Jacques Verstraëte,et al.  Proof Of A Conjecture Of Erdős On Triangles In Set-Systems , 2005, Comb..

[21]  B. Bollobás.  On generalized graphs , 1965 .