Multiple cross-intersecting families of signed sets

A k-signed r-set on[n] = {1, ..., n} is an ordered pair (A, f), where A is an r-subset of [n] and f is a function from A to [k]. Families A1, ..., Ap are said to be cross-intersecting if any set in any family Ai intersects any set in any other family Aj. Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [n]. Our aim is to generalise Hilton's bound to one for families of k-signed r-sets on [n]. The main tool developed is an extension of Katona's cyclic permutation argument.peer-reviewe

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

Computation of maximal projection constants

The linear projection constant $\Pi(E)$ of a finite-dimensional real Banach space $E$ is the smallest number $C\in [0,+\infty)$ such that $E$ is a $C$-absolute retract in the category of real Banach spaces with bounded linear maps. We denote by $\Pi_n$ the maximal linear projection constant amongst $n$-dimensional Banach spaces. In this article, we prove that $\Pi_n$ may be determined by computing eigenvalues of certain two-graphs. From this result we obtain that the relative projection constants of codimension $n$ converge to $1+\Pi_n$. Furthermore, using the classification of $K_4$-free two-graphs, we give an alternative proof of $\Pi_2=\frac{4}{3}$. We also show by means of elementary functional analysis that for each integer $n\geq 1$ there exists a polyhedral $n$-dimensional Banach space $F_n$ such that $\Pi(F_n)=\Pi_n$.Comment: 27 page

Global hypercontractivity and its applications

The hypercontractive inequality on the discrete cube plays a crucial role in many fundamental results in the Analysis of Boolean functions, such as the KKL theorem, Friedgut's junta theorem and the invariance principle. In these results the cube is equipped with the uniform measure, but it is desirable, particularly for applications to the theory of sharp thresholds, to also obtain such results for general $p$-biased measures. However, simple examples show that when $p = o(1)$, there is no hypercontractive inequality that is strong enough. In this paper, we establish an effective hypercontractive inequality for general $p$ that applies to `global functions', i.e. functions that are not significantly affected by a restriction of a small set of coordinates. This class of functions appears naturally, e.g. in Bourgain's sharp threshold theorem, which states that such functions exhibit a sharp threshold. We demonstrate the power of our tool by strengthening Bourgain's theorem, thereby making progress on a conjecture of Kahn and Kalai and by establishing a $p$-biased analog of the invariance principle. Our results have significant applications in Extremal Combinatorics. Here we obtain new results on the Tur\'an number of any bounded degree uniform hypergraph obtained as the expansion of a hypergraph of bounded uniformity. These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in the area. In particular, we give general conditions under which the crosscut parameter asymptotically determines the Tur\'an number, answering a question of Mubayi and Verstra\"ete. We also apply the Junta Method to refine our asymptotic results and obtain several exact results, including proofs of the Huang--Loh--Sudakov conjecture on cross matchings and the F\"uredi--Jiang--Seiver conjecture on path expansions.Comment: Subsumes arXiv:1906.0556