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In this short note, we introduce cospectral graphons, paralleling the notion of cospectral graphs. As in the graph case, we give three equivalent definitions: by equality of spectra, by equality of cycle densities, and by a unitary transformation. We also give an example of two cospectral graphons that cannot be approximated by two sequences of cospectral graphs in the cut distance., Comment: 7 pages

It is well-known that if $(A,B)$ is an $\tfrac{\varepsilon}{2}$-regular pair (in the sense of Szemer\'edi) then there exist sets $A'\subset A$ and $B'\subset B'$ with $|A'|\leq \varepsilon|A|$ and $|B'|\leq \varepsilon|B|$ so that the degrees of all vertices in $A\setminus A'$ differ by at most $\varepsilon|B|$ and the degrees of all vertices in $B\setminus B'$ differ by at most $\varepsilon|A|$. We call such a property "$\varepsilon$-degularity". This leads to the notion of an "$\varepsilon$-degular" partition of a graph in the same way as the definition of $\varepsilon$-regular pairs leads to the notion of $\varepsilon$-regular partitions. We show that there exist graphs in which any $\varepsilon$-degular partition requires the number of clusters to be $\mathrm{tower}(\Theta(\varepsilon^{-1/3}))$. That is, even though degularity is a substantial relaxation of regularity, in general one cannot improve much on the bounds that come with Szemer\'edi's regularity lemma., Comment: 13 pages, 1 figure

In their study of the giant component in inhomogeneous random graphs, Bollob\'as, Janson, and Riordan introduced a class of branching processes parametrized by a possibly unbounded graphon. We prove that two such branching processes have the same distribution if and only if the corresponding graphons are fractionally isomorphic, a notion introduced by Greb\'ik and Rocha. A different class of branching processes was introduced by Hladk\'y, Nachmias, and Tran in relation to uniform spanning trees in finite graphs approximating a given connected graphon. We prove that two such branching processes have the same distribution if and only if the corresponding graphons are fractionally isomorphic up to scalar multiple. Combined with a recent result of Archer and Shalev, this implies that if uniform spanning trees of two dense graphs have a similar local structure, they have a similar scaling limit. As a side result we give a characterization of fractional isomorphism for graphs as well as graphons in terms of their connected components., Comment: 44 pages, 1 figure; minor changes compared to the previous version

A theorem of Frieze from 1985 asserts that the total weight of the minimum spanning tree of the complete graph $K_n$ whose edges get independent weights from the distribution $UNIFORM[0,1]$ converges to Ap\'ery's constant in probability, as $n\to\infty$. We generalize this result to sequences of graphs $G_n$ that converge to a graphon $W$. Further, we allow the weights of the edges to be drawn from different distributions (subject to moderate conditions). The limiting total weight $\kappa(W)$ of the minimum spanning tree is expressed in terms of a certain branching process defined on $W$, which was studied previously by Bollob\'as, Janson and Riordan in connection with the giant component in inhomogeneous random graphs., Comment: 20 pages, 1 figure

Each graphon $W:\Omega^2\rightarrow[0,1]$ yields an inhomogeneous random graph model $G(n,W)$. We show that $G(n,W)$ is asymptotically almost surely connected if and only if (i) $W$ is a connected graphon and (ii) the measure of elements of $\Omega$ of $W$-degree less than $\alpha$ is $o(\alpha)$ as $\alpha\rightarrow 0$. These two conditions encapsulate the absence of several linear-sized components, and of isolated vertices, respectively. We study in bigger detail the limit probability of the property that $G(n,W)$ contains an isolated vertex, and, more generally, the limit distribution of the minimum degree of $G(n,W)$., Comment: 22 pages, 3 figures

Greb\'ik and Rocha [Fractional Isomorphism of Graphons, Combinatorica 42, pp 365-404 (2022)] extended the well studied notion of fractional isomorphism of graphs to graphons. We prove that fractionally isomorphic graphons can be approximated in the cut distance by fractionally isomorphic finite graphs. This answers the main question from ibid. As an easy but convenient corollary, we deduce that every regular graphon can be approximated by regular graphs., Comment: 23 pages, 2 figures; as will appear in European Journal of Combinatorics

The theory of limits of permutations leads to limit objects called permutons, which are certain Borel measures on the unit square. We prove that permutons avoiding a given permutation of order $k$ have a particularly simple structure. Namely, almost every fiber of the disintegration of the permuton (say, along the x-axis) consists only of atoms, at most $(k-1)$ many, and this bound is sharp. We use this to give a simple proof of the `permutation removal lemma'., Comment: 17 pages, 4 figures. Reorganization of the previous version of the paper by incorporating the appendix into the main body. Addition of relevant citations and contextualization

Flip processes, introduced in [Garbe, Hladk\'y, \v{S}ileikis, Skerman: From flip processes to dynamical systems on graphons], are a class of random graph processes defined using a rule which is just a function $\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labelled graphs of a fixed order $k$ into itself. The process starts with an arbitrary given $n$-vertex graph $G_0$. In each step, the graph $G_i$ is obtained by sampling $k$ random vertices $v_1,\ldots,v_k$ of $G_{i-1}$ and replacing the induced graph $G_{i-1}[v_1,\ldots,v_k]$ by $\mathcal{R}(G_{i-1}[v_1,\ldots,v_k])$. Using the formalism of dynamical systems on graphons associated to each such flip process from ibid. we study several specific flip processes, including the triangle removal flip process and its generalizations, 'extremist flip processes' (in which $\mathcal{R}(H)$ is either a clique or an independent set, depending on whether $e(H)$ has less or more than half of all potential edges), and 'ignorant flip processes' in which the output $\mathcal{R}(H)$ does not depend on $H$.

We introduce a class of random graph processes, which we call flip processes. Each such process is given by a rule which is a function $\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labeled $k$-vertex graphs into itself ($k$ is fixed). The process starts with a given $n$-vertex graph $G_0$. In each step, the graph $G_i$ is obtained by sampling $k$ random vertices $v_1,\ldots,v_k$ of $G_{i-1}$ and replacing the induced graph $F:=G_{i-1}[v_1,\ldots,v_k]$ by $\mathcal{R}(F)$. This class contains several previously studied processes including the Erd\H{o}s--R\'enyi random graph process and the triangle removal process. Actually, our definition of flip processes is more general, in that $\mathcal{R}(F)$ is a probability distribution on $\mathcal{H}_k$, thus allowing randomised replacements. Given a flip process with a rule $\mathcal{R}$, we construct time-indexed trajectories $\Phi:\mathcal{W}_0\times [0,\infty)\rightarrow\mathcal{W}_0$ in the space of graphons. We prove that for any $T > 0$ starting with a large finite graph $G_0$ which is close to a graphon $W_0$ in the cut norm, with high probability the flip process will stay in a thin sausage around the trajectory $(\Phi(W_0,t))_{t=0}^T$ (after rescaling the time by the square of the order of the graph). These graphon trajectories are then studied from the perspective of dynamical systems. Among others topics, we study continuity properties of these trajectories with respect to time and initial graphon, existence and stability of fixed points and speed of convergence (whenever the infinite time limit exists). We give an example of a flip process with a periodic trajectory.

We prove that there is $c>0$ such that for all sufficiently large $n$, if $T_1,\dots,T_n$ are any trees such that $T_i$ has $i$ vertices and maximum degree at most $cn/\log n$, then $\{T_1,\dots,T_n\}$ packs into $K_n$. Our main result actually allows to replace the host graph $K_n$ by an arbitrary quasirandom graph, and to generalize from trees to graphs of bounded degeneracy that are rich in bare paths, contain some odd degree vertices, and only satisfy much less stringent restrictions on their number of vertices., Comment: 157 pages, 4 figures; small improvements throughout compared to the previous version

We develop a limit theory of Latin squares, paralleling the recent limit theories of dense graphs and permutations. We introduce a notion of density, an appropriate version of the cut distance, and a space of limit objects - so-called Latinons. Key results of our theory are the compactness of the limit space and the equivalence of the topologies induced by the cut distance and the left-convergence. Last, using Keevash's recent results on combinatorial designs, we prove that each Latinon can be approximated by a finite Latin square., Comment: 66 pages, 1 figure, final version published in Discrete Analysis

For a graph $H$, its homomorphism density in graphs naturally extends to the space of two-variable symmetric functions $W$ in $L^p$, $p\geq e(H)$, denoted by $t(H,W)$. One may then define corresponding functionals $\|W\|_{H}:=|t(H,W)|^{1/e(H)}$ and $\|W\|_{r(H)}:=t(H,|W|)^{1/e(H)}$ and say that $H$ is (semi-)norming if $\|.\|_{H}$ is a (semi-)norm and that $H$ is weakly norming if $\|.\|_{r(H)}$ is a norm. We obtain two results that contribute to the theory of (weakly) norming graphs. Firstly, answering a question of Hatami, who estimated the modulus of convexity and smoothness of $\|.\|_{H}$, we prove that $\|.\|_{r(H)}$ is not uniformly convex nor uniformly smooth, provided that $H$ is weakly norming. Secondly, we prove that every graph $H$ without isolated vertices is (weakly) norming if and only if each component is an isomorphic copy of a (weakly) norming graph. This strong factorisation result allows us to assume connectivity of $H$ when studying graph norms. In particular, we correct an error in the original statement of the aforementioned theorem by Hatami., Comment: 10 pages

Given a measurable set $A\subset \mathbb R^d$ we consider the "large-distance graph" $\mathcal{G}_A$, on the ground set $A$, in which each pair of points from $A$ whose distance is bigger than 2 forms an edge. We consider the problems of maximizing the $2d$-dimensional Lebesgue measure of the edge set as well as the $d$-dimensional Lebesgue measure of the vertex set of a large-distance graph in the $d$-dimensional Euclidean space that contains no copies of a complete graph on $k$ vertices. The former problem may be seen as a continuous analogue of Tur\'an's classical graph theorem, and the latter as a graph-theoretic analogue of the classical isodiametric problem. Our main result yields an analogue of Mantel's theorem for large-distance graphs. Our approach employs an isodiametric inequality in an annulus, which might be of independent interest., Comment: 15 pages, 3 figure; minor edits including more details in the proof of Theorem 1.8

The theory of graphons comes with a natural sampling procedure, which results in an inhomogeneous variant of the Erd\H{o}s--R\'enyi random graph, called $W$-random graphs. We prove, via the method of moments, a limit theorem for the number of $r$-cliques in such random graphs. We show that, whereas in the case of dense Erd\H{o}s--R\'enyi random graphs the fluctuations are normal of order $n^{r-1}$, the fluctuations in the setting of $W$-random graphs may be of order $0, n^{r-1}$, or $n^{r-0.5}$. Furthermore, when the fluctuations are of order $n^{r-0.5}$ they are normal, while when the fluctuations are of order $n^{r-1}$ they exhibit either normal or a particular type of chi-square behavior whose parameters relate to spectral properties of $W$. These results can also be deduced from a general setting [Janson and Nowicki, PTRF 1991], based on the projection method. In addition to providing alternative proofs, our approach makes direct links to the theory of graphons., Comment: 22 pages, 3 figures; accepted to Journal of Graph Theory

The theory of graphons comes with the so-called cut norm and the derived cut distance. The cut norm is finer than the weak* topology (when considering the predual of $L^{1}$-functions). Dole\v{z}al and Hladk\'y [J. Combin. Theory Ser. B 137 (2019), 232-263] showed, that given a sequence of graphons, a cut distance accumulation graphon can be pinpointed in the set of weak* accumulation points as a minimizer of the entropy. Motivated by this, we study graphon parameters with the property that their minimizers or maximizers identify cut distance accumulation points over the set of weak* accumulation points. We call such parameters cut distance identifying. Of particular importance are cut distance identifying parameters coming from homomorphism densities, $t(H,\cdot)$. This concept is closely related to the emerging field of graph norms, and the notions of the step Sidorenko property and the step forcing property introduced by Kr\'a\v{l}, Martins, Pach and Wrochna [J. Combin. Theory Ser. A 162 (2019), 34-54]. We prove that a connected graph is weakly norming if and only if it is step Sidorenko, and that if a graph is norming then it is step forcing. Further, we study convexity properties of cut distance identifying graphon parameters, and find a way to identify cut distance limits using spectra of graphons. We also show that continuous cut distance identifying graphon parameters have the {\guillemotleft}pumping property{\guillemotright}, and thus can be used in the proof of the Frieze-Kannan regularity lemma., Comment: 49 pages, 5 figures. Referees' comments incorporated. The most substantial change is a simplification of the proof of Proposition 2.15 (which does not rely on untrue (as we discovered) Exercise 4.18 from Lovasz's book anymore)

The theory of graphons is ultimately connected with the so-called cut norm. In this paper, we approach the cut norm topology via the weak* topology (when considering a predual of $L^{1}$-functions). We prove that a sequence $W_1,W_2,W_3,\ldots$ of graphons converges in the cut distance if and only if we have equality of the sets of weak* accumulation points and of weak* limit points of all sequences of graphons $W_1',W_2',W_3',\ldots$ that are weakly isomorphic to $W_1,W_2,W_3,\ldots$. We further give a short descriptive set theoretic argument that each sequence of graphons contains a subsequence with the property above. This in particular provides an alternative proof of the theorem of Lov\'asz and Szegedy about compactness of the space of graphons. We connect these results to "multiway cut" characterization of cut distance convergence from [Ann. of Math. (2) 176 (2012), no. 1, 151-219]. These results are more naturally phrased in the Vietoris hyperspace $K$ over graphons with the weak* topology. We show that graphons with the cut distance topology are homeomorphic to a closed subset of $K$, and deduce several consequences of this fact. From these concepts a new order on the space of graphons emerges. This order allows to compare how structured two graphons are. We establish basic properties of this "structurdness order"., Comment: 37 pages, 2 figures; added Sections 4.1 and 4.3, various fixes and improvements due to Ondrej Kurka and an anonymous referee

We study graphon counterparts of the chromatic and the clique number, the fractional chromatic number, the b-chromatic number, and the fractional clique number. We establish some basic properties of the independence set polytope in the graphon setting, and duality properties between the fractional chromatic number and the fractional clique number. We present a notion of perfect graphons and characterize them in terms of induced densities of odd cycles and its complements., Comment: 18 pages, accepted to European Journal of Combinatorics (special issue Eurocomb 2017)

Let $G$ be a connected graph in which almost all vertices have linear degrees and let $T$ be a uniform spanning tree of $G$. For any fixed rooted tree $F$ of height $r$ we compute the asymptotic density of vertices $v$ for which the $r$-ball around $v$ in $T$ is isomorphic to $F$. We deduce from this that if $\{G_n\}$ is a sequence of such graphs converging to a graphon $W$, then the uniform spanning tree of $G_n$ locally converges to a multi-type branching process defined in terms of $W$. As an application, we prove that in a graph with linear minimum degree, with high probability, the density of leaves in a uniform spanning tree is at least $1/e-o(1)$, the density of vertices of degree $2$ is at most $1/e+o(1)$ and the density of vertices of degree $k\geq 3$ is at most ${(k-2)^{k-2} \over (k-1)! e^{k-2}} + o(1)$. These bounds are sharp., Comment: 44 pages, error in Claim 4.1.3 fixed, as will appear in Journal of Statistical Physics, special issue devoted to Complex Networks

Given $D$ and $\gamma>0$, whenever $c>0$ is sufficiently small and $n$ sufficiently large, if $\mathcal{G}$ is a family of $D$-degenerate graphs of individual orders at most $n$, maximum degrees at most $\tfrac{cn}{\log n}$, and total number of edges at most $(1-\gamma)\binom{n}{2}$, then $\mathcal{G}$ packs into the complete graph $K_{n}$. Our proof proceeds by analysing a natural random greedy packing algorithm. This version of the manuscript corrects a small error that appeared in the published version [Adv Math, 354 (2019), 106739]., Comment: 48 pages. This version corrects small errors that we found after the publication in Adv Math. More details in Section 1.1

We prove that the accumulation points of a sequence of graphs $G_1,G_2,G_3,\ldots$ with respect to the cut-distance are exactly the weak$^*$ limit points of subsequences of the adjacency matrices (when all possible orders of the vertices are considered) that minimize the entropy over all weak$^*$ limit points of the corresponding subsequence. In fact, the entropy can be replaced by any map $W\mapsto \int\int f(W(x,y))$, where $f$ is a continuous and strictly concave function. Our proofs are elementary, and do not use the regularity lemma., Comment: 23 pages, 2 figures. Referees' comments incorporated

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labelled by using the numbers {1,2,...,n} in such a way that the absolute differences induced on the edges are pairwise distinct. We prove the following relaxation of the conjecture for each c>0 and for all n>n_0(c). Suppose that (i) the maximum degree of T is bounded by O(n/log n), and (ii) the vertex labels are chosen from the set {1,2,..., (1+c)n}. Then there is an injective labelling of V(T) such that the absolute differences on the edges are pairwise distinct. In particular, asymptotically almost all trees on n vertices admit such a labelling. As a consequence, for any such tree T we can pack (2+2c)n-1 copies of T into the complete graph of order (2+2c)n-1 cyclically. This proves an approximate version of the Ringel-Kotzig conjecture (which asserts the existence of a cyclic packing of 2n-1 copies of any T into the complete graph of order 2n-1) for these trees. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labelling with high probability., Comment: 46 pages, 3 figures; minor changes

Komlos [Komlos: Tiling Turan Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum-degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph H, thus essentially extending the Hajnal-Szemeredi theorem which deals with the case when H is a clique. We give a proof of a graphon version of Komlos's theorem. To prove this graphon version, and also to deduce from it the original statement about finite graphs, we use the machinery introduced in [Hladky, Hu, Piguet: Tilings in graphons, arXiv:1606.03113]. We further prove a stability version of Komlos's theorem., Comment: 21 pages, 1 figure; accepted to Journal of Graph Theory

Hladky, Hu, and Piguet [Tilings in graphons, preprint] introduced the notions of matching and fractional vertex covers in graphons. These are counterparts to the corresponding notions in finite graphs. Combinatorial optimization studies the structure of the matching polytope and the fractional vertex cover polytope of a graph. Here, in analogy, we initiate the study of the structure of the set of all matchings and of all fractional vertex covers in a graphon. We call these sets the matching polyton and the fractional vertex cover polyton. We also study properties of matching polytons and fractional vertex cover polytons along convergent sequences of graphons. As an auxiliary tool of independent interest, we prove that a graphon is $r$-partite if and only if it contains no graph of chromatic number $r+1$. This in turn gives a characterization of bipartite graphons as those having a symmetric spectrum., Comment: 32 pages, 2 figures; more background on graphons and analysis, a new section on spectra of bipartite graphons, numerous corrections based on suggestion by an anonymous referee

We introduce a counterpart to the notion of vertex disjoint tilings by copy of a fixed graph F to the setting of graphons. The case F=K_2 gives the notion of matchings in graphons. We give a transference statement that allows us to switch between the finite and limit notion, and derive several favorable properties, including the LP-duality counterpart to the classical relation between the fractional vertex covers and fractional matchings/tilings, and discuss connections with property testing. As an application of our theory, we determine the asymptotically almost sure F-tiling number of inhomogeneous random graphs \mathbb{G}(n,W). As another application, in an accompanying paper [Hladky, Hu, Piguet: Komlos's tiling theorem via graphon covers, preprint] we give a proof of a strengthening of a theorem of Komlos [Komlos: Tiling Tur\'an Theorems, Combinatorica, 2000]., Comment: 25 pages, 5 figures; to appear in European Journal of Combinatorics

In this short note we observe that recent results of Abert and Hubai and of Csikvari and Frenkel about Benjamini--Schramm continuity of the holomorphic moments of the roots of the chromatic polynomial extend to the theory of dense graph sequences. We offer a number of problems and conjectures motivated by this observation., Comment: 9 pages

The theory of dense graph limits comes with a natural sampling process which yields an inhomogeneous variant G(n,W) of the Erdos-Renyi random graph. Here we study the clique number of these random graphs. We establish the concentration of the clique number of G(n,W) for each fixed n, and give examples of graphons for which G(n,W) exhibits wild long-term behavior. Our main result is an asymptotic formula which gives the almost sure clique number of these random graphs. We obtain a similar result for the bipartite version of the problem. We also make an observation that might be of independent interest: Every graphon avoiding a fixed graph is countably-partite., Comment: 30 pages, 2 figure; accepted to publication in Random Structures and Algorithms

We prove that among all flag triangulations of manifolds of odd dimension 2r-1 with sufficiently many vertices the unique maximizer of the entries of the f-, h-, g- and gamma-vector is the balanced join of r cycles. Our proof uses methods from extremal graph theory., Comment: Clarifications and new references, title has changed

In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemer\'edi regularity lemma: we decompose the graph $G$, find a suitable combinatorial structure inside the decomposition, and then embed the tree $T$ into $G$ using this structure. Since for sparse graphs $G$, the decomposition given by the regularity lemma is not helpful, we use a more general decomposition technique. We show that each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. In this paper, we introduce this novel decomposition technique. In the three follow-up papers, we find a combinatorial structure suitable inside the decomposition, which we then use for embedding the tree., Comment: 41 pages, 6 figures; further referees' comments incorporated, the most substantial of which being a newly written Section 3.8

This is the second of a series of four papers in which we prove the following relaxation of the Loebl-Komlos--Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics; this decomposition might be viewed as an analogue of a regular partition for sparse graphs. In the present paper, we find a combinatorial structure inside this decomposition. In the last two papers, we refine the structure and use it for embedding the tree $T$., Comment: 38 pages, 4 figures; new is Section 5.1.1; accepted to SIDMA

This is the last paper of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number~$k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first two papers of this series, we decomposed the host graph $G$, and found a suitable combinatorial structure inside the decomposition. In the third paper, we refined this structure, and proved that any graph satisfying the conditions of the above approximate version of the Loebl-Komlos-Sos Conjecture contains one of ten specific configurations. In this paper we embed the tree $T$ in each of the ten configurations., Comment: 81 pages, 12 figures. A fix reflecting the change of Preconfiguration Clubs in Paper III, additional small changes

This is the third of a series of four papers in which we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $\alpha>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+\alpha)n$ vertices of degree at least $(1+\alpha)k$ contains each tree $T$ of order $k$ as a subgraph. In the first paper of the series, we gave a decomposition of the graph $G$ into several parts of different characteristics. In the second paper, we found a combinatorial structure inside the decomposition. In this paper, we will give a refinement of this structure. In the forthcoming fourth paper, the refined structure will be used for embedding the tree $T$., Comment: 59 pages, 4 figures; further comments by a referee incorporated; this includes a subtle but important fix to Lemma 5.1; as a consequence, Preconfiguration Clubs was changed

Loebl, Koml\'os and S\'os conjectured that every $n$-vertex graph $G$ with at least $n/2$ vertices of degree at least $k$ contains each tree $T$ of order $k+1$ as a subgraph. We give a sketch of a proof of the approximate version of this conjecture for large values of $k$. For our proof, we use a structural decomposition which can be seen as an analogue of Szemer\'edi's regularity lemma for possibly very sparse graphs. With this tool, each graph can be decomposed into four parts: a set of vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibiting certain expansion properties. We then exploit the properties of each of the parts of $G$ to embed a given tree $T$. The purpose of this note is to highlight the key steps of our proof. Details can be found in [arXiv:1211.3050].

We prove that for any pair of constants $\epsilon>0$ and $\Delta$ and for $n$ sufficiently large, every family of trees of orders at most $n$, maximum degrees at most $\Delta$, and with at most $\binom{n}{2}$ edges in total packs into $K_{(1+\epsilon)n}$. This implies asymptotic versions of the Tree Packing Conjecture of Gyarfas from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof., Comment: 38 pages, 2 figures; suggestions by an anonymous referee incorporated; accepted to Israel J Math

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique extremal graph. For $(r-1)|M|

We find, for all sufficiently large $n$ and each $k$, the maximum number of edges in an $n$-vertex graph which does not contain $k+1$ vertex-disjoint triangles. This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corradi-Hajnal Theorem., Comment: 41 pages (including 11 pages of appendix), 4 figures, 2 tables

We prove the following version of the Loebl-Komlos-Sos Conjecture: For every alpha>0 there exists a number M such that for every k>M every n-vertex graph G with at least (0.5+alpha)n vertices of degree at least (1+alpha)k contains each tree T of order k as a subgraph. The method to prove our result follows a strategy common to approaches which employ the Szemeredi Regularity Lemma: we decompose the graph G, find a suitable combinatorial structure inside the decomposition, and then embed the tree T into G using this structure. However, the decomposition given by the Regularity Lemma is not of help when G is sparse. To surmount this shortcoming we use a more general decomposition technique: each graph can be decomposed into vertices of huge degree, regular pairs (in the sense of the Regularity Lemma), and two other objects each exhibiting certain expansion properties., Comment: 166 pages, 18 figures, 2 tables. This version should now be consider obsolate and is replaced by a series: [arXiv:1408.3858], [arXiv:1408.3871], [arXiv:1408.3866], [arXiv:1408.3870]. The only change compared to the previous arXiv version is this comment

S.Janson [Poset limits and exchangeable random posets, Combinatorica 31 (2011), 529--563] defined limits of finite posets in parallel to the emerging theory of limits of dense graphs. We prove that each poset limit can be represented as a kernel on the unit interval with the standard order, thus answering an open question of Janson. We provide two proofs: real-analytic and combinatorial. The combinatorial proof is based on a Szemeredi-type Regularity Lemma for posets which may be of independent interest. Also, as a by-product of the analytic proof, we show that every atomless ordered probability space admits a measure-preserving and almost order-preserving map to the unit interval., Comment: Final version accepted by Transactions of Amer Math Soc, 18 pages

We characterize f-vectors of sufficiently large three-dimensional flag Gorenstein* complexes, essentially confirming a conjecture of Gal [Discrete Comput. Geom., 34 (2), 269--284, 2005]. In particular, this characterizes f-vectors of large flag triangulations of the 3-sphere. Actually, our main result is more general and describes the structure of closed flag 3-manifolds which have many edges. Looking at the 1-skeleta of these manifolds we reduce the problem to a certain question in extremal graph theory. We then resolve this question by employing the Supersaturation Theorem of Erdos and Simonovits., Comment: Trans. AMS, to appear

A graph H is called common if the total number of copies of H in every graph and its complement asymptotically minimizes for random graphs. A former conjecture of Burr and Rosta, extending a conjecture of Erdos asserted that every graph is common. Thomason disproved both conjectures by showing that the complete graph of order four is not common. It is now known that in fact the common graphs are very rare. Answering a question of Sidorenko and of Jagger, Stovicek and Thomason from 1996 we show that the 5-wheel is common. This provides the first example of a common graph that is not three-colorable., Comment: 9 pages

Using the formalism of flag algebras, we prove that every triangle-free graph $G$ with $n$ vertices contains at most $(n/5)^5$ cycles of length five. Moreover, the equality is attained only when $n$ is divisible by five and $G$ is the balanced blow-up of the pentagon. We also compute the maximal number of pentagons and characterize extremal graphs in the non-divisible case provided $n$ is sufficiently large. This settles a conjecture made by Erd\H{o}s in 1984., Comment: 16 pages, accepted to Journal of Combinatorial Theory Ser. A

This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turan's theorem, which deals with graphs G=([n],E) such that no member of the restriction set consisting of all r-tuples on [n] induces a copy of K_r. Firstly, we examine what happens when this restriction set is replaced just by all r-tuples touching a given m-element set. That is, we determine the maximal number of edges in an n-vertex such that no K_r hits a given vertex set. Secondly, we consider sparse random restriction sets. An r-uniform hypergraph R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if for any graph G on [n] with more edges than the Turan number ex(n,K_r) (respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G. We determine the thresholds for random r-uniform hypergraphs to be Turannical and to epsilon-Turannical. Thirdly, we transfer this result to sparse random graphs, using techniques recently developed by Schacht [Extremal results for random discrete structures] to prove the Kohayakawa-Luczak-Rodl Conjecture on Turan's theorem in random graphs., Comment: 33 pages, minor improvements thanks to two referees

A famous conjecture of Lov\'asz states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle., Comment: 26 pages, 3 figures; referees' comments incorporated; accepted for publication in Journal of Combinatorial Theory, series B

For a fixed bipartite graph H and given number c, 0

Motivated by the Caccetta-Haggkvist Conjecture, we prove that every digraph on n vertices with minimum outdegree 0.3465n contains an oriented triangle. This improves the bound of 0.3532n of Hamburger, Haxell and Kostochka. The main new tool we use in our proof is the theory of flag algebras developed recently by Razborov., Comment: 19 pages, 7 figures; this is the final version to appear in Combinatorica

Much of extremal graph theory has concentrated either on finding very small subgraphs of a large graph (Turan-type results) or on finding spanning subgraphs (Dirac-type results). In this paper we are interested in finding intermediate-sized subgraphs. We investigate minimum degree conditions under which a graph G contains squared paths and squared cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of G is large. This extends results of Fan and Kierstead [J. Combin. Theory Ser. B 63 (1995), 55--64] and of Komlos, Sarkozy, and Szemeredi [Random Structures Algorithms 9 (1996), 193--211] concerning the containment of a spanning squared path and a spanning squared cycle, respectively. Our results show that such minimum degree conditions constitute not merely an interpolation between the corresponding Turan-type and Dirac-type results, but exhibit other interesting phenomena., Comment: 44 pages, 2 figures; small errors corrected

We give a proof of Brooks' theorem and its list coloring extension using the algebraic method of Alon and Tarsi; this also shows that the Brooks' theorem remains valid in a more general game coloring setting., Comment: Brooks' Theorem extended to a game coloring setting, author list changed; 5 pages

We prove that for all epsilon>0 there are c>0 and n_0 such that for all n>n_0 the following holds. For any two-colouring of the edges of $K_{n,n,n}$ one colour contains copies of all trees T of order t<(3-epsilon)n/2 and with maximum degree at most n^c. This confirms a conjecture of Schelp., Comment: 25 pages

Let s

We prove a version of the Loebl-Komlos-Sos Conjecture for dense graphs. For each q>0 there exists a number $n_0\in \mathbb{N}$ such that for any n>n_0 and k>qn the following holds: if G be a graph of order n with at least n/2 vertices of degree at least k, then any tree of order k+1 is a subgraph of G., Comment: 56 pages, 8 figures; substantial changes as suggested by a referee