Let F be a family of pairs of sets. We call it an (a, b)-set system if for every set-pair (A,B) in F we have that |A| = a, |B| = b, A ∩ B = ∅. The following classical result on families of cross-intersecting set-pairs is due to Bollobás [6]. Let F be an (a, b)-set system with the cross-intersecting property, i.e., for (Ai, Bi), (Aj, Bj) ∈ F with i 6= j we have that both Ai ∩ Bj and Aj ∩ Bi are ...

Let Σk consist of all k-graphs with three edges D1, D2, D3 such that |D1∩D2| = k − 1 and D1 △ D2 ⊆ D3. The exact value of the Turán function ex(n,Σk) was computed for k = 3 by Bollobás [Discrete Math., 8 (1974) 21–24] and for k = 4 by Sidorenko [Math Notes, 41 (1987) 247–259]. Let the k-graph Tk ∈ Σk have edges {1, . . . , k}, {1, 2, . . . , k − 1, k + 1}, and {k, k + 1, . . . , 2k − 1}. Frankl...

Let R = ( d t ) , h be a Riordan array, where ∑ n ≥ 0 and . We show that if the matrix [ ⋯ 1 2 ⋮ ⋱ ] is totally positive, then so array

Gyarmati, Matolcsi, and Ruzsa recently noted [2] that Han-type inequalities can be applied to sumsets in much the same way that they can be applied to characteristic functions of sets of random variables (the usual situation). While it is not true (in general) that sumsets satisfy a logsubmodular relation in an obvious way, it is natural to ask whether they permit a weaker property, by way of f...

Mantel’s theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let K− 4 = {123, 124, 134}, F6 = {123, 124, 345, 156} and F = {K− 4 , F6}: for n 6= 5 the unique F-free 3-graph of order n and maximum size is the balanced complete tripartite 3-graph S3(n) (for n ...

For k ≥ 2 and l ≥ 0, let C (k, l) denote the family of all cycles of length l modulo k, and C (k) := C (l, 0). In proving the Even Cycle Theorem, we determine almost tight bounds on ex(n,C (k, l)) as well as a method for giving upper bounds on ex(n,C ) for various other infinite families of cycles. Bollobás [2] was the first to show that ex(n,C (k, l)) is linear in n for each pair (k, l) such t...

The inactivation of cytoplasmic malate dehydrogenase (L-malate: NAD+ oxidoreductase, EC 1.1.1.37) from porcine heart and the specific modification of arginyl residues have been found to occur when the enzyme is inhibited with the reagent butanedione in sodium borate buffer. The inactivation of the enzyme was found to follow pseudo-first order kinetics. This loss of enzymatic activity was concom...