The parity search problem

We prove that for any positive integers $n$ and $d$ there exists a collection consisting of $f=d\log n+O(1)$ subsets $A_1, A_2, \ldots, A_f$ of $[n]$ such that for any two distinct subsets $X$ and $Y$ of $[n]$ whose size is at most $d$ there is an index $i\in [f]$ for which $| A_i\cap X|$ and $|A_i\cap Y|$ have different parity. Here we think of $d$ as fixed whereas $n$ is thought of as tending to infinity, and the base of the logarithm is $2$. Translated into the language of combinatorial search theory, this tells us that $$d \log n+O(1)$$ queries suffice to identify up to $d$ marked items from a totality of $n$ items if the answers one gets are just whether an even or an odd number of marked elements has been queried, even if the search is performed non-adaptively. Since the entropy method easily yields a matching lower bound for the adaptive version of this problem, our result is asymptotically best possible. This answers a question posed by D\'aniel Gerbner and Bal\'azs Patk\'os in Gyula O.H. Katona's Search Theory Seminar at the R\'enyi institute

The Clique Density Theorem

Tur\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geq 2$ every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually adjacent vertices. The corresponding extremal graphs are balanced $(r-1)$-partite graphs. The question as to how many such $r$-cliques appear at least in any $n$-vertex graph with $\gamma n^2$ edges has been intensively studied in the literature. In particular, Lov\'{a}sz and Simonovits conjectured in the 1970s that asymptotically the best possible lower bound is given by the complete multipartite graph with $\gamma n^2$ edges in which all but one vertex class is of the same size while the remaining one may be smaller. Their conjecture was recently resolved for $r=3$ by Razborov and for $r=4$ by Nikiforov. In this article, we prove the conjecture for all values of $r$.Comment: 25 pages, second version addresses changes arising from the referee report

Counting odd cycles in locally dense graphs

We prove that for any given $\varepsilon>0$ and $d\in [0,1]$, every sufficiently large $(\varepsilon, d)$-dense graph $G$ contains for each odd integer $r$ at least $(d^r-\varepsilon)|V(G)|^r$ cycles of length $r$. Here, $G$ being $(\varepsilon, d)$-dense means that every set $X$ containing at least~$\varepsilon\,|V(G)|$ vertices spans at least $\tfrac d2\, |X|^2$ edges, and what we really count is the number of homomorphisms from an $r$-cycle into $G$. The result adresses a question of Y. Kohayakawa, B. Nagle, V. R\"odl, and M. Schacht

Maximum star densities

Given an integer $k \geq 2$ and a real number $\gamma\in [0, 1]$, which graphs of edge density $\gamma$ contain the largest number of $k$-edge stars? For $k=2$ Ahlswede and Katona proved that asymptotically there cannot be more such stars than in a clique or in the complement of a clique (depending on the value of $\gamma$). Here we extend their result to all integers $k\ge 2$.Comment: third version addresses changes arising from the referee report

Decomposing graphs into forests

We give a simple graph-theoretic proof of a classical result due to C. St. J. A. Nash-Williams on covering graphs by forests. Moreover we derive a slight generalisation of this statement where some edges are preassigned to distinct forests.Comment: This version differs slightly from the version published by the Journa

Forcing quasirandomness with triangles

We study forcing pairs for quasirandom graphs. Chung, Graham, and Wilson initiated the study of families $\\mathcal F$ of graphs with the property that if a large graph $G$ has approximately homomorphism density $p^{e(F)}$ for some fixed $p\in(0,1]$ for every $F\in \mathcal F$, then $G$ is quasirandom with density $p$. Such families $\mathcal F$ are said to be forcing. Several forcing families were found over the last three decades and characterising all bipartite graphs $F$ such that $(K_2,F)$ is a forcing pair is a well-known open problem in the area of quasirandom graphs, which is closely related to Sidorenko's conjecture. In fact, most of the known forcing families involve bipartite graphs only. We consider forcing pairs containing the triangle $K_3$. In particular, we show that if $(K_2,F)$ is a forcing pair, then so is $(K_3,F')$, where $F'$ is obtained from $F$ by replacing every edge of $F$ by a triangle (each of which introduces a new vertex). For the proof we first show that $(K_3,C'_4)$ is a forcing pair, which strengthens related results of Simonovits and S\'os and of Conlon et al.Comment: 16 pages, second version addresses changes arising from the referee report

Weighted variants of the Andr\'asfai-Erd\H{o}s-S\'os Theorem

A well known result due to Andr\'asfai, Erd\H{o}s, and S\'os asserts that for $r\ge 2$ every $K_{r+1}$-free graph on $n$ vertices with $\delta(G)>\frac{3r-4}{3r-1}n$ is $r$-partite. We study related questions in the context of weighted graphs, which are motivated by recent work on the Ramsey-Tur\'an problem for cliques.Comment: 8 figures; revised according to referee repor

A tale of stars and cliques

We show that for an infinitely many natural numbers $k$ there are $k$-uniform hypergraphs which admit a `rescaling phenomenon' as described in [9]. More precisely, let $\mathcal{A}(k,I, n)$ denote the class of $k$-graphs on $n$ vertices in which the sizes of all pairwise intersections of edges belong to a set $I$. We show that if $k=rt^2$ for some $r\ge 1$ and $t\ge 2$, and~$I$ is chosen in some special way, the densest graphs in $\mathcal{A}(rt^2,I, n)$ are either dominated by stars of large degree, or basically, they are `$t$-thick' $rt^2$-graphs in which vertices are partitioned into groups of $t$ vertices each and every edge is a union of $tr$ such groups. It is easy to see that, unlike in stars, the maximum degree of $t$-thick graphs is of a lower order than the number of its edges. Thus, if we study the graphs from $\mathcal{A}(rt^2,I, n)$ with a prescribed number of edges $m$ which minimize the maximum degree, around the value of $m$ which is the number of edges of the largest $t$-thick graph, a rapid, discontinuous phase transition can be observed. Interestingly, these two types of $k$-graphs determine the structure of all hypergraphs in $\mathcal{A}(rt^2,I, n)$. Namely, we show that each such hypergraph can be decomposed into a $t$-thick graph $H_T$, a special collection $H_S$ of stars, and a sparse `left-over' graph $H_R$.Comment: second version addresses changes arising from the referee report

The chromatic number of finite type-graphs

By a finite type-graph we mean a graph whose set of vertices is the set of all $k$-subsets of $[n]=\{1,2,\ldots, n\}$ for some integers $n\ge k\ge 1$, and in which two such sets are adjacent if and only if they realise a certain order type specified in advance. Examples of such graphs have been investigated in a great variety of contexts in the literature with particular attention being paid to their chromatic number. In recent joint work with Tomasz {\L}uczak, two of the authors embarked on a systematic study of the chromatic numbers of such type-graphs, formulated a general conjecture determining this number up to a multiplicative factor, and proved various results of this kind. In this article we fully prove this conjecture.Comment: second version addresses changes arising from the referee report

Hypergraphs with vanishing Tur\'an density in uniformly dense hypergraphs

P. Erd\H{o}s [On extremal problems of graphs and generalized graphs, Israel Journal of Mathematics 2 (1964), 183-190] characterised those hypergraphs $F$ that have to appear in any sufficiently large hypergraph $H$ of positive density. We study related questions for $3$-uniform hypergraphs with the additional assumption that $H$ has to be uniformly dense with respect to vertex sets. In particular, we characterise those hypergraphs $F$ that are guaranteed to appear in large uniformly dense hypergraphs $H$ of positive density. We also review the case when the density of the induced subhypergraphs of $H$ may depend on the proportion of the considered vertex sets.Comment: 26 pages, this version addresses changes arising from the referee report