Powers of Hamilton cycles of high discrepancy are unavoidable

The P\'osa-Seymour conjecture asserts that every graph on $n$ vertices with minimum degree at least $(1 - 1/(r+1))n$ contains the $r^{th}$ power of a Hamilton cycle. Koml\'os, S\'ark\"ozy and Szemer\'edi famously proved the conjecture for large $n.$ The notion of discrepancy appears in many areas of mathematics, including graph theory. In this setting, a graph $G$ is given along with a $2$-coloring of its edges. One is then asked to find in $G$ a copy of a given subgraph with a large discrepancy, i.e., with many more edges in one of the colors. For $r \geq 2,$ we determine the minimum degree threshold needed to find the $r^{th}$ power of a Hamilton cycle of large discrepancy, answering a question posed by Balogh, Csaba, Pluh\'ar and Treglown. Notably, for $r \geq 3,$ this threshold approximately matches the minimum degree requirement of the P\'osa-Seymour conjecture

Weighted Tur\'an theorems with applications to Ramsey-Tur\'an type of problems

We study extensions of Tur\'an Theorem in edge-weighted settings. A particular case of interest is when constraints on the weight of an edge come from the order of the largest clique containing it. These problems are motivated by Ramsey-Tur\'an type problems. Some of our proofs are based on the method of graph Lagrangians, while the other proofs use flag algebras. Using these results, we prove several new upper bounds on the Ramsey-Tur\'an density of cliques. Other applications of our results are in a recent paper of Balogh, Chen, McCourt and Murley.Comment: 19 pages, 10 figure

Effective Bounds for Induced Size-Ramsey Numbers of Cycles

The induced size-Ramsey number r^indk(H) of a graph H is the smallest number of edges a (host) graph G can have such that for any k-coloring of its edges, there exists a monochromatic copy of H which is an induced subgraph of G. In 1995, in their seminal paper, Haxell, Kohayakawa and Łuczak showed that for cycles, these numbers are linear for any constant number of colours, i.e., r^indk(Cn)≤Cn for some C=C(k). The constant C comes from the use of the regularity lemma, and has a tower type dependence on k. In this paper we significantly improve these bounds, showing that r^indk(Cn)≤O(k102)n when n is even, thus obtaining only a polynomial dependence of C on k. We also prove r^indk(Cn)≤eO(klogk)n for odd n, which almost matches the lower bound of eΩ(k)n. Finally, we show that the ordinary (non-induced) size-Ramsey number satisfies r^k(Cn)=eO(k)n for odd n. This substantially improves the best previous result of eO(k2)n, and is best possible, up to the implied constant in the exponent. To achieve our results, we present a new host graph construction which, roughly speaking, reduces our task to finding a cycle of approximate given length in a graph with local sparsity

Ramsey numbers of hypergraphs of a given size

The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erd\H{o}s and Graham asked to maximize the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed $k \ge 3$ and $q \ge 2$ we prove that the largest possible $q$-color Ramsey number of a $k$-uniform hypergraph with $m$ edges is at most $\mathrm{tw}_k(O(\sqrt{m})),$ where $\mathrm{tw}$ denotes the tower function. We also present a construction showing that this bound is tight for $q \ge 4$. This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for $k \geq 4$ and the lower bound for $k=3$. Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs

Large cliques or co-cliques in hypergraphs with forbidden order-size pairs

The well-known Erd\H{o}s-Hajnal conjecture states that for any graph $F$, there exists $\epsilon>0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^{\epsilon}$. We consider a variant of the Erd\H{o}s-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$, for every $S \subseteq \{0,1,2,3,4\}$, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in $S$. In most cases the bounds are essentially tight. We also determine, for all $S$, whether the growth rate is polynomial or polylogarithmic. Some open problems remain.Comment: A preliminary version of this manuscript appeared as arXiv:2303.0957

Large cliques or cocliques in hypergraphs with forbidden order-size pairs

The well-known Erdős-Hajnal conjecture states that for any graph $F$, there exists $ϵ>0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^ϵ$. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0≤f≤(m2)$, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$, for every $S⊆{0,1,2,3,4}$, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in $S$. In most cases the bounds are essentially tight. We also determine, for all $S$, whether the growth rate is polynomial or polylogarithmic. Some open problems remain

Zbornik I. skupa hrvatske ranokršćanske arheologije (HRRANA)

Episkopalno središte, tri slavne nekropole Salone i Eufrazijeva bazilika u Poreču su najpoznatiji ranokršćanski lokaliteti u Hrvatskoj. Oni, međutim, nisu i jedini – uz njih su pronađeni brojni drugi, ne manje važni, ranokršćanski objekti, crkve, nekropole i pokretni nalazi. O njihovom značaju govori i činjenica da je više njih relevantno i u međunarodnom istraživanju određenih arheoloških tema. K tomu, nakon gotovo stoljeća i pol arheoloških iskopavanja, znanost uvijek iznova dolazi do novih otkrića. Takva poticajna situacija potaknula je znanstvenike sa Katedre za antičku provincijalnu i ranokršćansku arheologiju Odsjeka za arheologiju Filozofskog fakulteta u Zagrebu na organizaciju prvog nacionalnog ranokršćanskog arheološkog skupa. Slijedom toga je u Zagrebu od 15. do 17. ožujka 2018. održan prvi hrvatski skup ranokršćanske arheologije. Prvi cilj skupa, čiji akronim po početnim slovima naslova glasi HRRANA, je bio prezentiranje aktualnog stanja, i možebitno unaprjeđivanje, te važne znanstvene discipline. Objavom ovog zbornika radova nastoji se promovirati ne samo hrvatske ranokršćanske lokalitete, spomenike, arhitekturu, krajobraze, ikonografiju, epigrafiju i recentna arheološka istraživanja, nego i uputiti na različite istraživačke i metodološke probleme u istraživanju ranokršćanske arheologije u Hrvatskoj kroz rasprave, ispitivanja, znanstvena i praktična pitanja.Episkopalno središte, tri slavne nekropole Salone i Eufrazijeva bazilika u Poreču su najpoznatiji ranokršćanski lokaliteti u Hrvatskoj. Oni, međutim, nisu i jedini – uz njih su pronađeni brojni drugi, ne manje važni, ranokršćanski objekti, crkve, nekropole i pokretni nalazi. O njihovom značaju govori i činjenica da je više njih relevantno i u međunarodnom istraživanju određenih arheoloških tema. K tomu, nakon gotovo stoljeća i pol arheoloških iskopavanja, znanost uvijek iznova dolazi do novih otkrića. Takva poticajna situacija potaknula je znanstvenike sa Katedre za antičku provincijalnu i ranokršćansku arheologiju Odsjeka za arheologiju Filozofskog fakulteta u Zagrebu na organizaciju prvog nacionalnog ranokršćanskog arheološkog skupa. Slijedom toga je u Zagrebu od 15. do 17. ožujka 2018. održan prvi hrvatski skup ranokršćanske arheologije. Prvi cilj skupa, čiji akronim po početnim slovima naslova glasi HRRANA, je bio prezentiranje aktualnog stanja, i možebitno unaprjeđivanje, te važne znanstvene discipline. Objavom ovog zbornika radova nastoji se promovirati ne samo hrvatske ranokršćanske lokalitete, spomenike, arhitekturu, krajobraze, ikonografiju, epigrafiju i recentna arheološka istraživanja, nego i uputiti na različite istraživačke i metodološke probleme u istraživanju ranokršćanske arheologije u Hrvatskoj kroz rasprave, ispitivanja, znanstvena i praktična pitanja