The minimum number of triangular edges and a symmetrization method for multiple graphs

We give an asymptotic formula for the minimum number of edges contained in triangles in a graph having n vertices and e edges. Our main tool is a generalization of Zykov's symmetrization method that can be applied for several graphs simultaneously.Comment: Same paper, better presentation. Now it has 10 pages with 7 figure

The dimension of the region of feasible tournament profiles

Erd\H os, Lov\'asz and Spencer showed in the late 1970s that the dimension of the region of $k$-vertex graph profiles, i.e., the region of feasible densities of $k$-vertex graphs in large graphs, is equal to the number of non-trivial connected graphs with at most $k$ vertices. We determine the dimension of the region of $k$-vertex tournament profiles. Our result, which explores an interesting connection to Lyndon words, yields that the dimension is much larger than just the number of strongly connected tournaments, which would be the answer expected as the analogy to the setting of graphs

C5C_5 is almost a fractalizer

We determine the maximum number of induced copies of a 5-cycle in a graph on $n$ vertices for every $n$. Every extremal construction is a balanced iterated blow-up of the 5-cycle with the possible exception of the smallest level where for $n=8$, the M\"obius ladder achieves the same number of induced 5-cycles as the blow-up of a 5-cycle on 8 vertices. This result completes work of Balogh, Hu, Lidick\'y, and Pfender [Eur. J. Comb. 52 (2016)] who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method but we extend its use to small graphs.Comment: 24 page

Maximum Number of Almost Similar Triangles in the Plane

A triangle T′ is ε-similar to another triangle T if their angles pairwise differ by at most ε. Given a triangle T, ε\u3e0 and n∈N, Bárány and Füredi asked to determine the maximum number of triangles h(n,T,ε) being ε-similar to T in a planar point set of size n. We show that for almost all triangles T there exists ε=ε(T)\u3e0 such that h(n,T,ε)=n3/24(1+o(1)). Exploring connections to hypergraph Turán problems, we use flag algebras and stability techniques for the proof

Minimum number of edges that occur in odd cycles

If a graph has $n\ge4k$ vertices and more than $n^2/4$ edges, then it contains a copy of $C_{2k+1}$. In 1992, Erd\H{o}s, Faudree and Rousseau showed even more, that the number of edges that occur in a triangle is at least $2\lfloor n/2\rfloor+1$, and this bound is tight. They also showed that the minimum number of edges that occur in a $C_{2k+1}$ for $k\ge2$ is at least $11n^2/144-O(n)$, and conjectured that for any $k\ge2$, the correct lower bound should be $2n^2/9-O(n)$. Very recently, F\"uredi and Maleki constructed a counterexample for $k=2$ and proved asymptotically matching lower bound, namely that for any $\varepsilon>0$ graphs with $(1+\varepsilon)n^2/4$ edges contain at least $(2+\sqrt{2})n^2/16 \approx 0.2134n^2$ edges that occur in $C_5$. In this paper, we use a different approach to tackle this problem and obtain the following stronger result: Any $n$-vertex graph with at least $\lfloor n^2/4\rfloor+1$ edges has at least $(2+\sqrt{2})n^2/16-O(n^{15/8})$ edges that occur in $C_5$. Next, for all $k\ge 3$ and $n$ sufficiently large, we determine the exact minimum number of edges that occur in $C_{2k+1}$ for $n$-vertex graphs with more than $n^2/4$ edges, and show it is indeed equal to $\lfloor\frac{n^2}4\rfloor+1-\lfloor\frac{n+4}6\rfloor\lfloor\frac{n+1}6\rfloor=2n^2/9-O(n)$. For both results, we give a structural description of the extremal configurations as well as obtain the corresponding stability results, which answer a conjecture of F\"uredi and Maleki. The main ingredient is a novel approach that combines the flag algebras together with ideas from finite forcibility of graph limits. This approach allowed us to keep track of the extra edge needed to guarantee an existence of a $C_{2k+1}$. Also, we establish the first application of semidefinite method in a setting, where the set of tight examples has exponential size, and arises from different constructions

Minimizing the Number of Triangular Edges

We consider the problem of minimizing the number of edges that are contained in triangles, among n-vertex graphs with a given number of edges. For sufficiently large n, we prove an exact formula for this minimum, which partially resolves a conjecture of Füredi and Maleki

Combinatorics and Probability

For the past few decades, Combinatorics and Probability Theory have had a fruitful symbiosis, each benefitting from and influencing developments in the other. Thus to prove the existence of designs, probabilistic methods are used, algorithms to factorize integers need combinatorics and probability theory (in addition to number theory), and the study of random matrices needs combinatorics. In the workshop a great variety of topics exemplifying this interaction were considered, including problems concerning designs, Cayley graphs, additive number theory, multiplicative number theory, noise sensitivity, random graphs, extremal graphs and random matrices

Tilings and other combinatorial results

In this dissertation we treat three tiling problems and three problems in combinatorial geometry, extremal graph theory and sparse Ramsey theory. We first consider tilings of $\mathbb{Z}^n$. In this setting a tile $T$ is just a finite subset of $\mathbb{Z}^n$. We say that $T$ tiles $\mathbb{Z}^n$ if the latter set admits a partition into isometric copies of $T$. Chalcraft observed that there exist $T$ that do not tile $\mathbb{Z}^n$ but tile $\mathbb{Z}^{d}$ for some $d>n$. He conjectured that such $d$ exists for any given tile. We prove this conjecture in Chapter 2. In Chapter 3 we prove a conjecture of Lonc, stating that for any poset $P$ of size a power of $2$, if $P$ has a greatest and a least element, then there is a positive integer $k$ such that $[2]^k$ can be partitioned into copies of $P$. The third tiling problem is about vertex-partitions of the hypercube graph $Q_n$. Offner asked: if $G$ is a subgraph of $Q_n$ such $|G|$ is a power of $2$, must $V(Q_d)$, for some $d$, admit a partition into isomorphic copies of $G$? In Chapter 4 we answer this question in the affirmative. We follow up with a question in combinatorial geometry. A line in a planar set $P$ is a maximal collinear subset of $P$. P\'or and Wood considered colourings of finite $P$ without large lines with a bounded number of colours. In particular, they examined whether monochromatic lines always appear in such colourings provided that $|P|$ is large. They conjectured that for all $k,l \ge 2$ there exists an $n \ge 2$ such that if $|P| \ge n$ and $P$ does not contain a line of cardinality larger than $l$, then every colouring of $P$ with $k$ colours produces a monochromatic line. In Chapter 5 we construct arbitrarily large counterexamples for the case $k=l=3$. We follow up with a problem in extremal graph theory. For any graph, we say that a given edge is triangular if it forms a triangle with two other edges. How few triangular edges can there be in a graph with $n$ vertices and $m$ edges? For sufficiently large $n$ we prove a conjecture of F\"uredi and Maleki that gives an exact formula for this minimum. This proof is given in Chapter 6. Finally, Chapter 7 is concerned with degrees of vertices in directed hypergraphs. One way to prescribe an orientation to an $r$-uniform graph $H$ is to assign for each of its edges one of the $r!$ possible orderings of its elements. Then, for any $p$-set of vertices $A$ and any $p$-set of indices $I \subset [r]$, we define the $I$-degree of $A$ to be the number of edges containing vertices $A$ in precisely the positions labelled by $I$. Caro and Hansberg were interested in determining whether a given $r$-uniform hypergraph admits an orientation where every set of $p$ vertices has some $I$-degree equal to $0$. They conjectured that a certain Hall-type condition is sufficient. We show that this is true for $r$ large, but false in general.EPSR