Problems and Results on Additive Properties of General Sequences IV

Let A={a1,a2,⋯}, a1<a2<⋯, be an infinite sequence of positive integers. One defines its counting function as A(n)=Card{a∈A;a≤n} (n=0,1,2,⋯), and the representation functions R1(n), R2(n), R3(n) (n=0,1,2,⋯), as being the number of representations of n in the form: (1) n=a+a′, a∈A, a′∈A, (2) n=a+a′, a<a′, a∈A, a′∈A, (3) n=a+a′, a≤a′, a∈A, a′∈A, respectively. Of course, for any n≥0, R1(n)=R2(n)+R3(n), and R3(n) is equal either to R2(n)+1, if n is even and n/2 belongs to A, or to R2(n), otherwise. In the first three parts of this series of papers [Erdős and Sárközy, Part I, Pacific J. Math. 118 (1985), no. 2, 347–357; MR0789175 (86j:11015); Part II, Acta Math. Hungar. 48 (1986), 201–211; MR0858398 (88c:11016); Part III, the authors, Studia Sci. Math. Hungar. 22 (1987), no. 1, 53–63], regularity properties of the asymptotic behavior of the function R1 were studied. In Parts IV and V the authors study monotonicity properties of the three functions R1,R2,R3. In Part IV, they prove first that the function R1 is monotone increasing from a certain point on (i.e., there exists an n0 withR1(n+1)≥R1(n) for n≥n0) if and only if the sequence A contains all the integers from a certain point on. The proof uses elementary but complex considerations on counting functions. Secondly, they show that R2 has a different behavior, by exhibiting a class of sequences A satisfying A(n <n−cn1/3 for all large n and such that R2 is monotone increasing from some point onwards. The third result proved in Part IV is that if A(n)=o(n/logn) then the functions R2 and R3 cannot be monotone increasing from a certain point on. Here, the proof is based on analytic properties of the generating function f(z)=∑a∈Aza (|z|<1), corresponding to the sequence A. Part V treats the monotonicity of R3. The main result is as follows: If (4) limn→+∞(n−A(n))/logn=+∞, then lim supN→+∞∑k=1N(R3(2k)−R3(2k+1))=+∞. (Thus, roughly speaking, ai+aj assumes more even values than odd ones.) This theorem implies, firstly, that under hypothesis (4), which is weaker than A(n)=o(n/logn), R3 cannot be monotone increasing from a certain point on, and, secondly, that if A is an infinite "Sidon sequence'' (also called a "B2-sequence'', i.e., a sequence such that R3(n)≤1 for all n), then there are infinitely many integers k such that 2k can be represented in the form 2k=a+a′, a∈A, a′∈A, but 2k+1=a+a′, a∈A, a′∈A, is impossible. Part V finishes with the construction of a sequence showing that the main result is almost best possible. The proofs in Part V are of the same nature as those in Part IV

The critical window for the classical Ramsey-Tur\'an problem

The first application of Szemer\'edi's powerful regularity method was the following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.Comment: 34 page