Abstract. A graph on n vertices is said to be C-Ramsey if every clique or independent set of the graph has size at most C log n. The only known constructions of Ramsey graphs are probabilistic in nature, and it is generally believed that such graphs possess many of the same properties as dense random graphs. Here, we demonstrate one such property: for any fixed C > 0, every C-Ramsey graph on n vertices induces subgraphs of at least n 2−o(1) distinct sizes. This near-optimal result is closely related to two unresolved conjectures, the first due to Erdős and McKay and the second due to Erdős, Faudree and Sós, both from 1992.

An ordered graph H is a simple graph with a linear order on its vertex set. The corresponding Turán problem, first studied by Pach and Tardos, asks for the maximum number ex < (n, H) of edges in an ordered graph on n vertices that does not contain H as an ordered subgraph. It is known that ex < (n, H) > n 1+ε for some positive ε = ε(H) unless H is a forest that has a proper 2-coloring with one color class totally preceding the other one. Making progress towards a conjecture of Pach and Tardos, we prove that ex < (n, H) = n 1+o (1) holds for all such forests that are "degenerate" in a certain sense. This class includes every forest for which an n 1+o(1) upper bound was previously known, as well as new examples. Our proof is based on a density-increment argument.

A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell \geq r+1$ vertices $x_1,...,x_{\ell }$ such that all $r$-tuples $\{x_{i},x_{i+1},...,x_{i+r-1}\}$ (with subscripts modulo $\ell $) are edges of $\mathcal{H}$. An old problem of V. Sós, also posed independently by J. Verstraëte, asks for the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices, which has no tight cycle. Although this is a very basic question, until recently, no good upper bounds were known for this problem for $r\geq 3$. Here we prove that the answer is at most $n^{r-1+o(1)}$. This is tight up to the $o(1)$ error term, and it was shown recently by B. Janzer that this error term is indeed needed. Our proof is based on finding robust expanders in the line-graph of $\mathcal{H}$ together with certain density increment type arguments.

Let ω(G) and χ(G) denote the clique number and chromatic number of a graph G, respectively. The disjointness graph of a family of curves (continuous arcs in the plane) is the graph whose vertices correspond to the curves and in which two vertices are joined by an edge if and only if the corresponding curves are disjoint. A curve is called x-monotone if every vertical line intersects it in at most one point. An x-monotone curve is grounded if its left endpoint lies on the y-axis.We prove that if G is the disjointness graph of a family of grounded x-monotone curves such that ω(G) = k, then χ(G) ≤ k+12 . If we only require that every curve is x-monotone and intersects the y-axis, then we have χ(G) ≤ k+1 2 k+2 3 . Both of these bounds are best possible. The construction showing the tightness of the last result settles a 25 years old problem: it yields that there exist K k -free disjointness graphs of x-monotone curves such that any proper coloring of them uses at least Ω(k 4 ) colors. This matches the upper bound up to a constant factor. function f (k) = 4k 2 . (It is conjectured that the same is true with bounding function f (k) = O(k).) However, an ingenious construction of Burling [4] shows that the family of intersection graphs of axis-parallel boxes in R 3 is not χ-bounded. The χ-boundedness of intersection graphs of chords of a circle was established by Gyárfás [16,17]; see also Kostochka et al. [21,23]. Deciding whether a family of graphs is χ-bounded is often a difficult task [22].Computing the chromatic number of the disjointness graph of a family of objects, C, is equivalent to determining the clique cover number of the corresponding intersection graph G, that is, the minimum number of cliques whose vertices together cover the vertex set of G. This problem can be solved in polynomial time only for some very special families (for instance, if C consists of intervals along a line or arcs along a circle [15]). On the other hand, the problem is known to be NP-complete if C is a family of chords of a circle [18,14] or a family of unit disks in the plane [43,6], and in many other cases. There is a vast literature providing approximation algorithms or inapproximability results for the clique cover number [8,9]. Families of curves.A curve or string in R 2 is the image of a continuous function φ : [0, 1] → R d . A curve C ⊂ R 2 is called x-monotone if every vertical line intersects C in at most one point. We say that C is grounded at the curve L if one of the endpoints of C is in L, and this is the only intersection point of C and L. A grounded x-monotone curve is an x-monotone curve that is contained in the half-plane {x ≥ 0}, and whose left endpoint lies on the vertical line {x = 0}.It was first suggested by Erdős in the 1970s, and remained the prevailing conjecture for 40 years, that the family of intersection graphs of curves (the family of so-called "string graphs") is χ-bounded [3,24]. There were many promising facts pointing in this direction. Extending earlier results of McGuinness [31], Suk [42], and Lasoń ...

No abstract