Finite and Infinite Sets. Colloquia Mathematica Societatis by A. Hajnal, L. Lovász, V. T. Sós

By A. Hajnal, L. Lovász, V. T. Sós

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4. Let sé, @ be two AD families on N. The family sé is finer than uè (sé -< @) iff for each A G sé there is some B G & with A c B. 5. ). 6. Let sé be an AD family on N. We shall denote f+(sé)= [X ω}. ) A . R . D . M a t h i a s [13] introduced the notion of happy families and showed that f + (sé) is a happy family, if sé is a MAD family on N. We shall frequently use the following consequence of being happy and refer it as a "happiness" argument. If sé is a MAD family on N, {Xn: ne ω}^ S + (sé) and XQ 2 Χχ 2 .

T. G a l l a i for calling my REFERENCE [1] A . L i n d e n m a y e r , Mathematical models for cellular intemactions in development (I. Filaments with one-sided inputs), J. Theoret. BioL, 18(1968), 280-299. B . A n d r â s f ai Department of Mathematics, Faculty of Electrical Engineering, Technical University of Budapest, Budapest, H-1521, Hungary. -45 - C O L L O Q U I A M A T H E M A T I C A S O C I E T A T I S JANOS B O L Y A 1 37. F I N I T E AND I N F I N I T E SETS, EGER ( H U N G A R Y ) , 1981.

Let G be a graph with n vertices, the vector c be a configuration to G, M be the adjacency matrix of G, E be a unit matrix, B = M + E and r{B) be the rank of B. Then A(T) is a cyclic cellular automaton if and only if Bx = c (mod 2) can be solved for arbitrary c, that is r{B) = n, namely in the case r(B) < n the vector c can be chosen independently from the column-vector of B. Hence one can easily see two following propositions. Proposition 3. A(G) has a Garden of Eden for every regular G of odd degree.

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