@MISC{Klazar94alinear, author = {Martin Klazar}, title = {A Linear Upper Bound in Extremal Theory of Sequences}, year = {1994} }

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Abstract

An extremal problem considering sequences related to Davenport-Schinzel sequences is investigated in this paper. We prove that f(x k ; n) = O(n) where the quantity on the left side is defined as the maximum length m of the sequence u = a 1 a 2 ::a m of integers such that 1) 1 a r n, 2) a r = a s ; r 6= s implies jr \Gamma sj k and 3) u contains no subsequence of the type x k (x stands for xx::x i-times).

...emerédi [Sz] to O(n log ∗ n) (for any of those functions, log ∗ n is the minimum number of 2’s in 22..2 making this tower greater or equal to n) but whether F = O(n) remained unclear. Hart and Sharir =-=[HS]-=- answered this question negatively: F = Θ(nα(n)) where α(n) is the functional inverse to the Ackermann function and goes to infinity but very slowly. Recently both sharp upper and lower bounds on the ...

...(nα(n)) where α(n) is the functional inverse to the Ackermann function and goes to infinity but very slowly. Recently both sharp upper and lower bounds on the functions f(ababab . . . , n) were found =-=[ASS]-=-, [S]. The aim of this paper is to give (linear) upper bounds for extremal functions of forbidden sequences a(i, k) = x i 1x i 2...x i k xi 1x i 2...x i k . Here xj are k distinct symbols and x i stan...

...ol sequences the results a i b i a i b i ∈ Lin ([AKV]) and ababa �∈ Lin ([HS]) yields the equivalence u ∈ Lin iff ababa �≤ u. This is not the case for a general sequence u because the construction in =-=[WS]-=- realizing the lower bound f(ababa, n) = Ω(n.α(n)) by segments in the plane proves also implicitly u1 = abcbadadbcd �∈ Lin (and ababa �≤ u1). Define the function f(n) = max{|v| | v ∈ Lin, �v� ≤ n, v i...

...e function f(u, n) = max{|v| | u �≤ v, �v� ≤ n, v is �u�-regular}. We shall show below that the maximum is defined correctly. The first problem considering f(u, n) was posed by Davenport and Schinzel =-=[DS]-=- in 1965 when they asked about the asymptotic growth of F = f(ababa, n) and in general of f(ababab . . . , n). They proved F = O(n log n/ log log n). This was later improved by Szemerédi [Sz] to O(n l...

...linear upper bound form the set Lin = {u | f(u, n) = O(n)} and our result may be reformulated as x i 1x i 2...x i k xi 1x i 2...x i k ∈ Lin. It generalizes the result ai b i a i b i ∈ Lin achieved in =-=[AKV]-=-. Finding all elements of Lin seems to be an interesting and not an easy problem (see concluding remarks). The linearity of a(i, k) is derived from two statements—Theorem A and Theorem B—which are per...