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For r geq 3 let d r(e) be the minimal d such that mathrm ex r(n, mathcal F )=o(n^2), where mathcal F is the family of r -uniform hypergraphs on d vertices with e edges. Prove that d r(e)=(r-…
Let G be a finite 3 -uniform hypergraph, and let F G( kappa) denote the collection of 3 -uniform hypergraphs with chromatic number kappa not containing G . If F G( aleph 1) is not empty then there exists $X …
Let K t (r) be the complete t -partite t -uniform hypergraph with r vertices in each class. Is it true that mathrm ex t(n,K t(r)) geq n^ t-r^ 1-t -o(1) for all t,r ?
Let t,k,r geq 2 . Let mathcal F be the family of all r -uniform hypergraphs with k vertices and s edges. Determine mathrm ex r(n, mathcal F ).
Let r geq 3 . There exists c r r^ -r such that, for any epsilon 0 , if n is sufficiently large, the following holds. Any r -uniform hypergraph on n vertices with at least (1+ epsilon)(n/r)^r many edges cont…
Let f(n;r,k) be the maximal number of edges in an r -uniform hypergraph which contains no set of k many independent edges. For all r geq 3 , f(n;r,k)= max left( binom rk-1 r , binom n r - binom n-k+1 r r…
Let m(n) be minimal such that there is an n -uniform hypergraph with m(n) edges which is 3 -chromatic. Estimate m(n) .
Let k geq 2 and A k subseteq [0,1] be the set of alpha such that there exists some beta( alpha) alpha with the property that, if G 1,G 2, ldots is a sequence of k -uniform hypergraphs with liminf frac…
Let r geq 2 and G be a r -uniform hypergraph with chromatic number 3 (that is, there is a 3 -colouring of the vertices of G such that no edge is monochromatic). Suppose any two edges of G have a non-empty in…
Does there exist a k 2 such that the k -sized subsets of 1, ldots,2k can be coloured with k+1 colours such that for every A subset 1, ldots,2k with lvert A rvert=k+1 all k+1 colours appear among the…
Let mathrm ex r(n;K r+1 ^r) be the maximum number of r -edges that can be placed on n vertices without forming a K r+1 ^r (the r -uniform complete graph on r+1 vertices). Is every r -hypergraph G on n …
Determine, for any k r 2 , the value of frac mathrm ex r(n,K k^r) binom n r , where mathrm ex r(n,K k^r) is the largest number of r -edges which can placed on n vertices so that there exists no set…
Let f(n;t) be minimal such that if a t -uniform hypergraph on n vertices contains at least f(n;t) edges then there must be four edges A,B,C,D such that A cup B= C cup D and A cap B=C cap D= emptyset.…
Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number aleph 0 .
Let R 3(n) be the minimal m such that if the edges of the 3 -uniform hypergraph on m vertices are 2 -coloured then there is a monochromatic copy of the complete 3 -uniform hypergraph on n vertices. Is there s…
Let F(n, alpha) denote the smallest m such that there exists a 2 -colouring of the edges of K n so that every X subseteq [n] with lvert X rvert geq m contains more than alpha binom lvert X rvert 2 many…
Let R r(n) denote the r -uniform hypergraph Ramsey number: the minimal m such that if we 2 -colour all edges of the complete r -uniform hypergraph on m vertices then there must be some monochromatic copy of the …
What is mathrm ex 3(n,K 4^3) ? That is, the largest number of 3 -edges which can placed on n vertices so that there exists no K 4^3 , a set of 4 vertices which is covered by all 4 possible 3 -edges.
Let alpha in[0,1/2) and n,t geq 1 . Let F^ (t) (n, alpha) be the smallest m such that we can 2 -colour the edges of the complete t -uniform hypergraph on n vertices such that if X subseteq [n] with $ lvert …