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Green developed an arithmetic regularity lemma to prove a strengthening of Roth's theorem on arithmetic progressions in dense sets. It states that for every ϵ>0 there is some N0(ϵ) such that for every N≥N0 (ϵ) and A ⊂ [N] with ∣A∣=αN, there is some nonzero d such that A contains at least (α3-ϵ)N three-term arithmetic progressions with common difference d. We prove that the minimum N0(ϵ) in Green's theorem is an exponential tower of twos of height on the order of log(1/ϵ). Both the lower and upper bounds are new. This shows that the tower-type bounds that arise from the use of a regularity lemma in this application are quantitatively necessary. [ABSTRACT FROM AUTHOR]

Prior research indicates that making choices before a painful task can sometimes reduce pain. We examined the possibility that independent and interdependent self-construals moderate the effect of choice on pain. Further, we tested between two types of choice: instrumental and non-instrumental. Healthy normotensive undergraduates were randomly assigned to one of three conditions prior to the cold pressor task. Participants in an instrumental choice condition selected which hand to immerse in the water and were told this choice might help reduce their pain. Non-instrumental choice participants selected which hand to immerse but were given no information about potential pain reduction. Control participants were given no choice or additional instructions. Low interdependence individuals reported less pain than high interdependence individuals-but only when given an instrumental choice. These data indicate that not all forms of choice reduce pain and not all individuals benefit from choice. Instead, individuals low in interdependence exhibit pain relief from instrumental choices., Author(s): Jacob Fox[sup.1] , Shane R. Close[sup.1] , Jason P. Rose[sup.1] , Andrew L. Geers[sup.1] Author Affiliations: (1) University of Toledo, Toledo, OH, USA Introduction In contemporary medical care, there [...]

The clique removal lemma says that for every r≥3 $r\ge 3$ and ε>0 $\varepsilon \gt 0$, there exists some δ>0 $\delta \gt 0$ so that every n $n$‐vertex graph G $G$ with fewer than δnr $\delta {n}^{r}$ copies of Kr ${K}_{r}$ can be made Kr ${K}_{r}$‐free by removing at most εn2 $\varepsilon {n}^{2}$ edges. The dependence of δ $\delta $ on ε $\varepsilon $ in this result is notoriously difficult to determine: it is known that δ−1 ${\delta }^{-1}$ must be at least super‐polynomial in ε−1 ${\varepsilon }^{-1}$, and that it is at most of tower type in log ε−1 $\mathrm{log} {\varepsilon }^{-1}$. We prove that if one imposes an appropriate minimum degree condition on G $G$, then one can actually take δ $\delta $ to be a linear function of ε $\varepsilon $ in the clique removal lemma. Moreover, we determine the threshold for such a minimum degree requirement, showing that above this threshold we have linear bounds, whereas below the threshold the bounds are once again super‐polynomial, as in the unrestricted removal lemma. We also investigate this question for other graphs besides cliques, and prove some general results about how minimum degree conditions affect the bounds in the graph removal lemma. [ABSTRACT FROM AUTHOR]