By Noga Alon (auth.), Imre Bárány, József Solymosi, Gábor Sági (eds.)

Szemerédi's impression on trendy arithmetic, in particular in combinatorics, additive quantity thought, and theoretical desktop technology, is big. This quantity is a party of Szemerédi's achievements and character, at the social gathering of his 70th birthday. It exemplifies his striking imaginative and prescient and particular state of mind. a couple of colleagues and pals, all best professionals of their fields, have contributed their most up-to-date learn papers to this quantity. the themes contain extension and functions of the regularity lemma, the life of k-term mathematics progressions in numerous subsets of the integers, extremal difficulties in hypergraphs thought, and random graphs, them all attractive, Szemerédi variety arithmetic. It additionally comprises released bills of the 1st , very unique and hugely profitable Polymath tasks, one led through Tim Gowers and the opposite by means of Terry Tao.

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The work of Marx [33], improving earlier ideas of Grohe [26] shows that in fact, for every graph H, the treewidth of H essentially captures the complexity of this problem . More precisely, this means that if the Exponential Time Hypothesis of [29] holds, that is, 3-SAT on m variables cannot be solved in time 2o(m ) , then there is no algorithm that solves the colored H -subgraph problem on an n vertex graph in time n O ( W / log w), where H is a fixed graph and w = w(H) is its treewidth. Note that, as usual, the little-o notation here means that formally one has to consider an infinite family of graphs H , and the term o(wI log w) is a quantity whose ratio to ui] logw tends to zero as w tends to infinity.

Note that the construction in [4] provides H(k, n)-universal graphs with n vertices , but their number of edges exceeds that of the graphs constructed in [5] by a logarithmic factor. • The results of Grohe [26] and Marx [33], described in Section 6 apply to general binary Constraint Satisfaction Problems (CSPs, for short), showing that if the naturally defined graph corresponding to a general binary CSP has treewidth w, then, assuming the Exponential Time Hypothesis of [29], there is no algorithm that solves the problem in time dO( W / log w), where d is the size of the domain of each variable of the CSP problem .

1) even for the simplest families of subsets. (4) As I already said above, we can make the vague term "1 - c part of all billard paths" in Theorem 1 precise by using the product measure on the set of all initial conditions of the billiard paths. Since the initial condition consists of a starting point y E [0, 1)2 and an initial direction (angle) e E [0,27r) , the natural measure here is simply the product of the two-dimensional Lebesgue measure on the unit square and the normalized one-dimensional Lebesgue measure .