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Register a new userIntersection theorems for triangles
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Added by
Andrey Kupavskii
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In particular, we show that if a set $P$ of $n$ points is in convex position, then the largest intersecting family of triangles with vertices in $P$ contains at most $(frac{1}{4}+o(1))binom{n}{3}$ triangles.
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Let $r(n,k)$ (resp. $s(n,k)$) be the number of Schroder paths (resp. little Schroder paths) of length $2n$ with $k$ hills, and set $r(0,0)=s(0,0)=1$. We bijectively establish the following recurrence relations: begin{align*} r(n,0)&=sumlimits_{j=0}^{n-1}2^{j}r(n-1,j), r(n,k)&=r(n-1,k-1)+sumlimits_{j=k}^{n-1}2^{j-k}r(n-1,j),quad 1le kle n, s(n,0) &=sumlimits_{j=1}^{n-1}2cdot3^{j-1}s(n-1,j), s(n,k) &=s(n-1,k-1)+sumlimits_{j=k+1}^{n-1}2cdot3^{j-k-1}s(n-1,j),quad 1le kle n. end{align*} The infinite lower triangular matrices $[r(n,k)]_{n,kge 0}$ and $[s(n,k)]_{n,kge 0}$, whose row sums produce the large and little Schroder numbers respectively, are two Riordan arrays of Bell type. Hence the above recurrences can also be deduced from their $A$- and $Z$-sequences characterizations. On the other hand, it is well-known that the large Schroder numbers also enumerate separable permutations. This propelled us to reveal the connection with a lesser-known permutation statistic, called initial ascending run, whose distribution on separable permutations is shown to be given by $[r(n,k)]_{n,kge 0}$ as well.
Alon and Yuster proved that the number of orientations of any $n$-vertex graph in which every $K_3$ is transitively oriented is at most $2^{lfloor n^2/4rfloor}$ for $n geq 10^4$ and conjectured that the precise lower bound on $n$ should be $n geq 8$. We confirm their conjecture and, additionally, characterize the extremal families by showing that the balanced complete bipartite graph with $n$ vertices is the only $n$-vertex graph for which there are exactly $2^{lfloor n^2/4rfloor}$ such orientations.
The study of intersection problems in Extremal Combinatorics dates back perhaps to 1938, when Paul ErdH{o}s, Chao Ko and Richard Rado proved the (first) `ErdH{o}s-Ko-Rado theorem on the maximum possible size of an intersecting family of $k$-element subsets of a finite set. Since then, a plethora of results of a similar flavour have been proved, for a range of different mathematical structures, using a wide variety of different methods. Structures studied in this context have included families of vector subspaces, families of graphs, subsets of finite groups with given group actions, and of course uniform hypergraphs with stronger or weaker intersection conditions imposed. The methods used have included purely combinatorial ones such as shifting/compressions, algebraic methods (including linear-algebraic, Fourier analytic and representation-theoretic), and more recently, analytic, probabilistic and regularity-type methods. As well as being natural problems in their own right, intersection problems have connections with many other parts of Combinatorics and with Theoretical Computer Science (and indeed with many other parts of Mathematics), both through the results themselves, and the methods used. In this survey paper, we discuss both old and new results (and both old and new methods), in the field of intersection problems. Many interesting open problems remain; we will discuss several. For expositional and pedagogical purposes, we also take this opportunity to give slightly streamlin
Given a graph sequence ${G_n}_{n geq 1}$ denote by $T_3(G_n)$ the number of monochromatic triangles in a uniformly random coloring of the vertices of $G_n$ with $c geq 2$ colors. This arises as a generalization of the birthday paradox, where $G_n$ corresponds to a friendship network and $T_3(G_n)$ counts the number of triples of friends with matching birthdays. In this paper we prove a central limit theorem (CLT) for $T_3(G_n)$ with explicit error rates. The proof involves constructing a martingale difference sequence by carefully ordering the vertices of $G_n$, based on a certain combinatorial score function, and using a quantitive version of the martingale CLT. We then relate this error term to the well-known fourth moment phenomenon, which, interestingly, holds only when the number of colors $c geq 5$. We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any $c geq 2$, which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of $T_3(G_n)$, whenever $c geq 5$. Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in Bhattacharya et al. (2017).
A convex geometric hypergraph or cgh consists of a family of subsets of a strictly convex set of points in the plane. There are eight pairwise nonisomorphic cghs consisting of two disjoint triples. These were studied at length by Bra{ss} (2004) and by Aronov, Dujmovic, Morin, Ooms, and da Silveira (2019). We determine the extremal functions exactly for seven of the eight configurations. The above results are about cyclically ordered hypergraphs. We extend some of them for triangle systems with vertices from a non-convex set. We also solve problems posed by P. Frankl, Holmsen and Kupavskii (2020), in particular, we determine the exact maximum size of an intersecting family of triangles whose vertices come from a set of $n$ points in the plane.
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