Some extremal problems on gamma-graphs

Let f(r)(n;k,s) denote the smallest t for which every r-graph with n vertices and t r-tuples contains a subgraph with k vertices and at least s r-tuples. It is proved that for integers k>r and s>1 there exists a positive constant ck,s such that f(r)(n;k,s)>ck,sn(rs−k)/(s−1). This inequality follows from a counting argument. Unfortunately a number of misprints make the proof seem incorrect. Inequality (1) ensures that there exists an r-graph H0(r) in M such that b(H0(r))<12m/(kr) (and not only m/(kr), as claimed on p. 60, 1.12). This, in turn, gives f(r)(n;k,s)≥12m, which is sufficient for the proof. The authors conjecture that limn→∞n−2f(3)(n;k,k−2) exists

Extremal Graph Theory: Basic Results

Η παρούσα διπλωματική εργασία έχει σκοπό να παρουσιάσει μία σφαιρική εικόνα της θεωρίας των ακραίων γραφημάτων, διερευνώντας κοινές τεχνικές και τον τρόπο που εφαρμόζονται σε κάποια από τα πιο διάσημα αποτελέσματα του τομέα. Το πρώτο κεφάλαιο είναι μία εισαγωγή στο θέμα και κάποιοι προαπαιτούμενοι ορισμοί και αποτελέσματα. Το δεύτερο κεφάλαιο αφορά υποδομές πυκνών γραφημάτων και εστιάζει σε σημαντικά αποτελέσματα όπως είναι το θεώρημα του Turán, το λήμμα κανονικότητας του Szemerédi και το θεώρημα των Erdős-Stone-Simonovits. Το τρίτο κεφάλαιο αφορά υποδομές αραιών γραφημάτων και ερευνά συνθήκες που εξαναγκάζουν ένα γράφημα που περιέχει ένα δοθέν έλασσον ή τοπολογικό έλασσον. Το τέταρτο και τελευταίο κεφάλαιο είναι μία εισαγωγή στην θεωρία ακραίων r-ομοιόμορφων υπεργραφημάτων και περιέχει αποτελέσματα που αφορούν συνθήκες οι οποίες τα εξαναγκάζουν να περιέχουν πλήρη r-γραφήματα και Χαμιλτονιανούς κύκλους.In this thesis, we take a general overview of extremal graph theory, investigating common techniques and how they apply to some of the more celebrated results in the field. The first chapter is an introduction to the subject and some preliminary definitions and results. The second chapter concerns substructures in dense graphs and focuses on important results such as Turán’s theorem, Szemerédi’s regularity lemma and the Erdős-Stone-Simonovits theorem. The third chapter concerns substructures in sparse graphs and investigates conditions which force a graph to contain a certain minor or topological minor. The fourth and final chapter is an introduction to the extremal theory of r-uniform hypergraphs and consists of a presentation of results concerning the conditions which force them to contain a complete r-graph and a Hamiltonian cycle

On the structure of graphs with forbidden induced substructures

One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints. In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs. Erdős and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every $2$-edge-colouring of the complete graph on $n$ vertices there is a monochromatic clique on at least $\frac{1}{2}\log n$ vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs. In the second part of this thesis we focus more on order-size pairs; an order-size pair $(n,e)$ is the family consisting of all graphs of order $n$ and size $e$, i.e. on $n$ vertices with $e$ edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs $(m,f)$, i.e. for $n$ approaching infinity, the limit superior of the fraction of all possible sizes $e$, such that the order-size pair $(n,e)$ does not avoid the pair $(m,f)$

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum