The Erdős-Ko-Rado properties of various graphs containing singletons

Let G=(V,E) be a graph. For r≥1, let be the family of independent vertex r-sets of G. For vV(G), let denote the star . G is said to be r-EKR if there exists vV(G) such that for any non-star family of pair-wise intersecting sets in . If the inequality is strict, then G is strictly r-EKR. Let Γ be the family of graphs that are disjoint unions of complete graphs, paths, cycles, including at least one singleton. Holroyd, Spencer and Talbot proved that, if GΓ and 2r is no larger than the number of connected components of G, then G is r-EKR. However, Holroyd and Talbot conjectured that, if G is any graph and 2r is no larger than μ(G), the size of a smallest maximal independent vertex set of G, then G is r-EKR, and strictly so if 2r<μ(G). We show that in fact, if GΓ and 2r is no larger than the independence number of G, then G is r-EKR; we do this by proving the result for all graphs that are in a suitable larger set Γ′Γ. We also confirm the conjecture for graphs in an even larger set Γ″Γ′

Invitation to intersection problems for finite sets

Extremal set theory is dealing with families, . F of subsets of an . n-element set. The usual problem is to determine or estimate the maximum possible size of . F, supposing that . F satisfies certain constraints. To limit the scope of this survey most of the constraints considered are of the following type: any . r subsets in . F have at least . t elements in common, all the sizes of pairwise intersections belong to a fixed set, . L of natural numbers, there are no . s pairwise disjoint subsets. Although many of these problems have a long history, their complete solutions remain elusive and pose a challenge to the interested reader.Most of the paper is devoted to sets, however certain extensions to other structures, in particular to vector spaces, integer sequences and permutations are mentioned as well. The last part of the paper gives a short glimpse of one of the very recent developments, the use of semidefinite programming to provide good upper bound

Extremális és véletlen struktúrák = Extremal and random structures

Résztvevők: T. Sós Vera, akadémikus, Szemerédi Endre, akadémikus, Füredi Zoltán akadémikus, Győri Ervin, a tudományok doktora, Elek Gábor a tudományok doktora, és témavezetőként Simonovits Miklós (akadémikus). Menetközben csatlakozott a pályázathoz Patkós Balázs. Itt, a rövid beszámolóban csak a legfontosabb témákat említem, Klasszikus Extremális és Ramsey problémák megoldása, ill. ezekkel rokon problémák. A Szemerédi Regularitási Lemma alkalmazásai, az extremális és Ramsey típusú kérdések kapcsolata, ezek kapcsolata a kvázivéletlenséggel, ""tulajdonság-teszteléssel"". Az extrém gráfelmélettel szoros kapcsolatban álló Erdős-Kleitman-Rothschild típusú tételek. A gráflimesz vizsgálata, alkalmazásai Hasonlóságok és különbségek a sűrű és ritka gráfok limesz-elméletében. ,,Sporadikus kérdések,'' pl. algebrai és geometriai alkalmazások. | Project leader: Miklós Simonovits Participants: Vera T. Sós , Endre Szemerédi, Zoltán Füredi, Ervin Győri, Gábor Elek. Balázs Patkós joined our group later. Here I have space only to mention the topics breafly. We were interested primarily in the connection, similarities and differences between deterministic and randomlike structures. Large part of our research was related to the Szemerédi Regularity Lemma and its various versions, and the applications of it, among others, in classical extremal graph and hypergraph problems. We also investigated the application of this lemma in quasi-randomness, property testing, and other related fields. We investigated the graph-limit theory, both for dense and veryy sparse graph sequences. Beside these, we investigated several ``Sporadic question,'' e.g. applications of our methods in algebra and geometry

Department of Mathematics Graduate Student Handbook

This is written for graduate students by graduate students, and should not be interpreted as a statement of official department policy