Let be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality || ≤ 2n−1. Suppose that ||=2n−1 + i. Choose the members of independently with probability p (delete them with probability 1 − p). The new family is intersecting with a certain probability. We try to maximize this probability by choosing appropriately. The exact maximum is determined in this paper for some small i. The analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all -element subsets. We determine the most probably inclusion-free family too, when the number of members is .

Given a locally finite connected infinite graph G, let the interval [pmin(G), pmax(G)] be the smallest interval such that if p > pmax(G), then every 1-independent bond percolation model on G with bond probability p percolates, and for p < pmin(G) none does. We determine this interval for trees in terms of the branching number of the tree. We also give some general bounds for other graphs G, in particular for lattices.

Let G = G(d) be a random graph with a given degree sequence d, such as a random r-regular graph where r ≥ 3 is fixed and n = |G| → ∞. We study the percolation phase transition on such graphs G, i.e., the emergence as p increases of a unique giant component in the random subgraph G[p] obtained by keeping edges independently with probability p. More generally, we study the emergence of a giant component in G(d) itself as d varies. We show that a single method can be used to prove very precise results below, inside and above the ‘scaling window’ of the phase transition, matching many of the known results for the much simpler model G(n, p). This method is a natural extension of that used by Bollobás and the author to study G(n, p), itself based on work of Aldous and of Nachmias and Peres; the calculations are significantly more involved in the present setting.

We show that the list chromatic number of a simple d-regular r-uniform hypergraph is at least (1/2r log(2r2) + o(1)) log d if d is large.

A family of sets (and the corresponding family of 0–1 vectors) is called t-cancellative if, for all distinct t + 2 members A1,. . ., At and B, C ∈ ,Let ct(n) be the size of the largest t-cancellative family on n elements, and let ct(n, r) denote the largest r-uniform family. We improve the previous upper bounds, e.g., we show c2(n) ≤ 20.322n (for n > n0). Using an algebraic construction we show that c2(n, 2k) = Θ(nk) for each k when n → ∞.

A rainbow subgraph of an edge-coloured graph is a subgraph whose edges have distinct colours. The colour degree of a vertex v is the number of different colours on edges incident with v. Wang and Li conjectured that for k ≥ 4, every edge-coloured graph with minimum colour degree k contains a rainbow matching of size at least ⌈k/2⌉. A properly edge-coloured K4 has no such matching, which motivates the restriction k ≥ 4, but Li and Xu proved the conjecture for all other properly coloured complete graphs. LeSaulnier, Stocker, Wenger and West showed that a rainbow matching of size ⌊k/2⌋ is guaranteed to exist, and they proved several sufficient conditions for a matching of size ⌈k/2⌉. We prove the conjecture in full.

Let HG mean that every s-colouring of E(H) produces a monochromatic copy of G in some colour class. Let the s-colour degree Ramsey number of a graph G, written RΔ(G; s), be min{Δ(H): HG}. If T is a tree in which one vertex has degree at most k and all others have degree at most ⌈k/2⌉, then RΔ(T; s) = s(k − 1) + ϵ, where ϵ = 1 when k is odd and ϵ = 0 when k is even. For general trees, RΔ(T; s) ≤ 2s(Δ(T) − 1).

To study sharpness of the upper bound, consider the double-starSa,b, the tree whose two non-leaf vertices have degrees a and b. If a ≤ b, then RΔ(Sa,b; 2) is 2b − 2 when a < b and b is even; it is 2b − 1 otherwise. If s is fixed and at least 3, then RΔ(Sb,b;s) = f(s)(b − 1) − o(b), where f(s) = 2s − 3.5 − O(s−1).

We prove several results about edge-colourings of bounded-degree graphs that are related to degree Ramsey numbers of paths. Finally, for cycles we show that RΔ(C2k + 1; s) ≥ 2s + 1, that RΔ(C2k; s) ≥ 2s, and that RΔ(C4;2) = 5. For the latter we prove the stronger statement that every graph with maximum degree at most 4 has a 2-edge-colouring such that the subgraph in each colour class has girth at least 5.

In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that if p(n) ≫ (log n/n)1/2, then asymptotically almost surely every subgraph of G(n, p) with minimum degree at least (2/3 + o(1))np contains a triangle packing that covers all but at most O(p−2) vertices. Moreover, the assumption on p is optimal up to the (log n)1/2 factor and the presence of the set of O(p−2) uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which says that if one imposes certain natural regularity conditions on all three pairs in a balanced 3-partite graph, then this graph contains a perfect triangle packing.

A subgraph of a hypergraph H is even if all its degrees are positive even integers, and b-bounded if it has maximum degree at most b. Let fb(n) denote the maximum number of edges in a linearn-vertex 3-uniform hypergraph which does not contain a b-bounded even subgraph. In this paper, we show that if b ≥ 12, thenfor some absolute constant B, thus establishing fb(n) up to polylogarithmic factors. This leaves open the interesting case b = 2, which is the case of 2-regular subgraphs. We are able to show for some constants c, C > 0 thatWe conjecture that f2(n) = n1 + o(1) as n → ∞.

R. H. Schelp conjectured that if G is a graph with |V(G)| = R(Pn, Pn) such that δ(G) > , then in every 2-colouring of the edges of G there is a monochromatic Pn. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree.

Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching–matching Ramsey number satisfying R(Sn, nK2, nK2) = 3n − 1. This extends R(nK2, nK2) = 3n − 1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp's conjecture in an asymptotic sense through a standard application of the Regularity Lemma.

It is sad that we are unable to hear Dick Schelp's reaction to our work generated by his conjecture.

Upper and lower bounds are proved for the maximum number of triangles in C2k+1-free graphs. The bounds involve extremal numbers related to appropriate even cycles.

Let 1 ≤ p ≤ r + 1, with r ≥ 2 an integer, and let G be a graph of order n. Let d(v) denote the degree of a vertex v ∈ V(G). We show that ifthen G has more than(r + 1)-cliques sharing a common edge. From this we deduce that ifthen G contains more thancliques of order r + 1.

In turn, this statement is used to strengthen the Erdős–Stone theorem by using ∑v ∈ V(G)dp(v) instead of the number of edges.

Richard Schelp completed his PhD in lattice theory in 1970 at Kansas State University. However, he did not take a traditional route to a PhD in mathematics and an outstanding career as a professor and a mathematical researcher. He grew up in rural northeast Missouri. He received his BS in mathematics and physics from the University of Central Missouri. After the completion of his master's degree in mathematics from Kansas State University, he assumed a position as an associate mathematician in the Applied Science Laboratory at Johns Hopkins University for five years. To start his PhD programme at Kansas State University, he had to quit a well-paying position. Also, he was already married to his wife Billie (Swopes) Schelp and he had a family – a daughter Lisa and a son Rick. This was a courageous step to take, but it says something about who Dick Schelp was.

In 1965 Erdős conjectured a formula for the maximum number of edges in a k-uniform n-vertex hypergraph without a matching of size s. We prove this conjecture for k = 3 and all s ≥ 1 and n ≥ 4s.

A family of sets is said to be intersecting if A ∩ B ≠ ∅ for all A, B ∈ . It is a well-known and simple fact that an intersecting family of subsets of [n] = {1, 2, . . ., n} can contain at most 2n−1 sets. Katona, Katona and Katona ask the following question. Suppose instead ⊂ [n] satisfies || = 2n−1 + i for some fixed i > 0. Create a new family p by choosing each member of independently with some fixed probability p. How do we choose to maximize the probability that p is intersecting? They conjecture that there is a nested sequence of optimal families for i = 1, 2,. . ., 2n−1. In this paper, we show that the families [n](≥r) = {A ⊂ [n]: |A| ≥ r} are optimal for the appropriate values of i, thereby proving the conjecture for this sequence of values. Moreover, we show that for intermediate values of i there exist optimal families lying between those we have found. It turns out that the optimal families we find simultaneously maximize the number of intersecting subfamilies of each possible order.

Standard compression techniques appear inadequate to solve the problem as they do not preserve intersection properties of subfamilies. Instead, our main tool is a novel compression method, together with a way of ‘compressing subfamilies’, which may be of independent interest.

We use the method of upper and lower solutions combined with degree-theoretic techniques to prove the existence of multiple positive solutions to some superlinear elliptic systems of the form

on a smooth, bounded domain Ω⊂ℝn with Dirichlet boundary conditions, under suitable conditions on g1 and g2. Our techniques apply generally to subcritical, superlinear problems with a certain concave–convex shape to their nonlinearity.

The abstract version of Struwe's monotonicity trick developed by Jeanjean and Toland for functionals depending on a real parameter is strengthened in the sense that it provides, for almost every value of the parameter, the existence of a bounded almost symmetric Palais–Smale sequence at the mountain-pass level whenever a mild symmetry assumption is set on the energy functional. In addition, the whole theory is extended to the case of continuous functionals on Banach spaces, in the framework of non-smooth critical point theory.

The Green function of singular limit-circle problems is constructed directly for the problem, not as a limit of sequences of regular Green's functions. This construction is used to obtain adjointness and self-adjointness conditions which are entirely analogous to the regular case. As an application, a new and explicit formula for the Green function of the classical Legendre problem is found.

Let (M, g) be a compact smooth N-dimensional Riemannian manifold without boundary. We consider the multiple existence of positive solutions of the problem

where Δg stands for the Laplacian in M and f ε C2(M).

We discuss a new class of Banach spaces which are wider than the strongly convex spaces introduced by Congxin Wu and Yongjin Li. We prove that the new class of Banach spaces lies strictly between either the class of uniformly convex spaces and strongly convex spaces or the class of fully k-convex spaces and strongly convex spaces. The new class of Banach spaces has inclusive relations with neither the class of locally uniformly convex spaces nor the class of nearly uniformly convex spaces. We obtain in addition some characterizations of this new class of Banach spaces.