By Mugnai D.

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**Additional info for Asymptotic behaviour, nodal lines and symmetry properties for solutions of superlinear elliptic equations near an eigenvalue**

**Example text**

At the end of a play Maker owns half of the edges (the red edges), so Maker’s graph (the red subgraph) must have degree mi ≥ di /2 in some vertex i (where di is the G-degree of i). Let Sur (G) be the largest integer S such that Maker can always force a red degree mi ≥ (di + S)/2, where di is the G-degree. 1) Sur (G) = max min max(mi − bi ), StrM StrB i meaning that, Sur (G) is the largest integer S such that, playing the Degree Game on G, Maker can always force a terminal lead ≥ S. That is, Maker has a strategy StrM with the property that, whatever strategy StrB is used by Breaker, at the end of the (StrM , StrB )-play there is always a vertex i where Maker’s degree mi is ≥ bi + S (here bi is Breaker’s degree in i).

B´ ar´ any, B. Doerr, Balanced partitions of vector sequences, Linear Alg. , 414 (2006), 464–469. [6] J. Beck, T. Fiala, Roth’s estimate of the discrepancy of integer sequences is nearly sharp, Combinatorica, 1 (1981), 319–325. [7] J. Beck, T. Fiala, “Integer-making” theorems, Discrete Appl. , 3 (1981), 1–6. [8] J. Beck, V. T. S´ os, Discrepancy theory, in: Handbook of combinatorics (ed. R. Graham, M. Gr¨ otschel, L. Lov´ asz), Elsevier, Amsterdam, 1995, 1405–1446. [9] V. Bergstr¨ om, Zwei S¨ atze u ¨ber ebene Vectorpolygone, Abh.

2 44 I. 1) 1 2 S(B) + E(B) implying S(B) < E(B) + 2η. Assume now that k > m. Then k1 ui = − nk+1 ui is outside S(B) − η B. Consequently, nk+1 ui is outside S(B) − η B as well. But the last sum is just the sum of the ﬁrst n − k elements of the sequence v1 , . . , vn that go with εi = −1. This sum is equal to 1 2 n−k n−k vi − 1 εi vi 1 ∈ 1 S(B) + E(B) B, 2 again. 1) in all cases. 1) holds for all η > 0, we have S(B) ≤ E(B). References [1] L. Babai, P. Frankl, Linear Algebra Methods in Combinatorics, Preliminary version 2.