Some results on the problem of Erdös and Hajnal.
Stability analysis for -wise intersecting families.
Standard monomials for q-uniform families and a conjecture of Babai and Frankl
Let n, k, α be integers, n, α>0, p be a prime and q=p α. Consider the complete q-uniform family We study certain inclusion matrices attached to F(k,q) over the field . We show that if l≤q−1 and 2l≤n then This extends a theorem of Frankl [7] obtained for the case α=1. In the proof we use arguments involving Gröbner bases, standard monomials and reduction. As an application, we solve a problem of Babai and Frankl related to the size of some L-intersecting families modulo q.
Strongly cancellative and recovering sets on lattices.
Subgraph densities in hypergraphs
Let r ≥ 2 be an integer. A real number α ∈ [0,1) is a jump for r if for any ε > 0 and any integer m ≥ r, any r-uniform graph with n > n₀(ε,m) vertices and density at least α+ε contains a subgraph with m vertices and density at least α+c, where c = c(α) > 0 does not depend on ε and m. A result of Erdös, Stone and Simonovits implies that every α ∈ [0,1) is a jump for r = 2. Erdös asked whether the same is true for r ≥ 3. Frankl and Rödl gave a negative answer by showing an infinite sequence...
The 11-element case of Frankl's conjecture.
The integer sequence A002620 and upper antagonistic functions.
The intersection structure of -intersecting families.
The Least Eigenvalue of Graphs whose Complements Are Uni- cyclic
A graph in a certain graph class is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum among all graphs in that class. Bell et al. have identified a subclass within the connected graphs of order n and size m in which minimizing graphs belong (the complements of such graphs are either disconnected or contain a clique of size n/2 ). In this paper we discuss the minimizing graphs of a special class of graphs of order n whose complements are connected and contains...
The structure of maximal zero-sum free sequences
The structure of maximum subsets of with no solutions to .
The Turán density of the hypergraph .
The Turán problem for hypergraphs on fixed size.
Three-and-more set theorems
In this paper we generalize classical 3-set theorem related to stable partitions of arbitrary mappings due to Erdős-de Bruijn, Katětov and Kasteleyn. We consider a structural generalization of this result to partitions preserving sets of inequalities and characterize all finite sets of such inequalities which can be preserved by a “small” coloring. These results are also related to graph homomorphisms and (oriented) colorings.
Towards a Katona type proof for the 2-intersecting Erdős-Ko-Rado theorem.
Traces of uniform families of sets.
Traces without maximal chains.
Triangle-intersecting families of graphs
Turán number of two vertex-disjoint copies of cliques
The Turán number of a given graph , denoted by , is the maximum number of edges in an -free graph on vertices. Applying a well-known result of Hajnal and Szemerédi, we determine the Turán number ) of a vertex-disjoint union of cliques and for all values of .
Unification of zero-sum problems, subset sums and covers of .