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We say a family of subsets of { 1 , 2 , … , n } is antipodal if it is closed under taking complements. We prove a best-possible isoperimetric inequality for antipodal families of subsets of { 1 , 2 , … , n } (of any size). Our inequality implies that for any k ∈ N , among all such families of size 2 k , a family consisting of the union of two antipodal ( k − 1 ) -dimensional subcubes has the smallest possible edge boundary. [ABSTRACT FROM AUTHOR]

An oriented 3-graph consists of a family of triples (3-sets), each of which is given one of its two possible cyclic orientations. A cycle in an oriented 3-graph is a positive sum of some of the triples that gives weight zero to each 2-set. Our aim in this paper is to consider the following question: how large can the girth of an oriented 3-graph be? We show that there exist oriented 3-graphs whose shortest cycle has length $$\frac{n^2}{2}(1+o(1))$$ : this is asymptotically best possible. We also show that there exist 3-tournaments whose shortest cycle has length $$\frac{n^2}{3}(1+o(1))$$ , in complete contrast to the case of 2-tournaments. [ABSTRACT FROM AUTHOR]

Abstract: How large can a family be if it does not contain with ? Our aim in this paper is to show that any such family has size at most . This is tight up to a multiplicative constant of 2. We also obtain similar results for families with , showing that they satisfy , where is a constant depending only on . [Copyright &y& Elsevier]

Abstract: Let be a family of subsets of an -set such that does not contain distinct sets and with . How large can be? Our aim in this note is to determine the maximum size of such an . This answers a question of Kalai. We also give some related results and conjectures. [Copyright &y& Elsevier]

Abstract: A finite set X in some Euclidean space is called Ramsey if for any k there is a d such that whenever is k-coloured it contains a monochromatic set congruent to X. This notion was introduced by Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. This question (made into a conjecture by Graham) has dominated subsequent work in Euclidean Ramsey theory. In this paper we introduce a new conjecture regarding which sets are Ramsey; this is the first ever ‘rival’ conjecture to the conjecture above. Calling a finite set transitive if its symmetry group acts transitively—in other words, if all points of the set look the same—our conjecture is that the Ramsey sets are precisely the transitive sets, together with their subsets. One appealing feature of this conjecture is that it reduces (in one direction) to a purely combinatorial statement. We give this statement as well as several other related conjectures. We also prove the first non-trivial cases of the statement. Curiously, it is far from obvious that our new conjecture is genuinely different from the old. We show that they are indeed different by proving that not every spherical set embeds in a transitive set. This result may be of independent interest. [Copyright &y& Elsevier]

Abstract: A k-signed r-set on is an ordered pair , where A is an r-subset of and f is a function from A to . Families are said to be cross-intersecting if any set in any family intersects any set in any other family . Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of . Our aim is to generalise Hilton''s bound to one for families of k-signed r-sets on . The main tool developed is an extension of Katona''s cyclic permutation argument. [Copyright &y& Elsevier]

Our aim in this note is to show that if A is a family of (k/3) 3-sets then there are at least (k/4) 3-sets then there are at least (k/4) 4-sets that may be written as unions of two 2-sets in the shadow of A. This is the first non-trivial case of a more general conjecture about unions of shadows. [Copyright &y& Elsevier]

We show that for any k,m,p,c, if G is a Kk-free graph on N then there is an independent set of vertices in G that contains an (m,p,c)-set. Hence if G is a Kk-free graph on N, then one can solve any partition regular system of equations in an independent set. This is a common generalization of partition regularity theorems of Rado (who characterized systems of linear equations Ax=0 a solution of which can be found monochromatic under any finite coloring of N) and Deuber (who provided another characterization in terms of (m,p,c)-sets and a partition theorem for them), and of Ramsey''s theorem itself. [Copyright &y& Elsevier]

Abstract: A function f from to is said to be feebly continuous at a point if there exist sequences and with . Dales asked if every function has a point of feeble continuity. Our aim in this short note is to show that (assuming the Continuum Hypothesis) the answer is ‘no’. Dales also asked what happens for functions taking only two values: we show that in this case the answer is ‘yes’. [Copyright &y& Elsevier]