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It is proved that, for an arbitrary field K not of characteristic 2 and arbitrary \( n \ge 2 \), if functions f : K→K and h : K→K satisfy f [x1, . . . , xn] = h (x1 + . . . + xn) whenever x1, . . . , xn are distinct elements of K, then f is equal to a polynomial of degree at most n over K. (Here f [x1, . . . , xn] denotes the divided difference of f at the distinct points x1, . . . , xn.) The case of a field of characteristic 2 is also considered.

The smallest (nontrivial) identity graph is known to have six points and the smallest identity tree seven. It is now shown that the smallest cubic identity graphs have 12 points and that exactly two of them are planar, namely those constructed by Frucht in his proof that every finite group is isomorphic to the automorphism group of some cubic graph. Both of these graphs can be obtained from plane trees by joining consecutive endpoints; it is shown that when applied to identity trees this construction leads to identity graphs except in certain specified cases. In appendices all connected cubic graphs with 10 points or fewer, and all cubic nonseparable planar graphs with 12 points, are displayed.

A set is said to be σ-finite (with respect to Λs-measure) if it can be expressed as a countable sum of sets of finite Λs-measure. I have proved(1) that every non σ-finite analytic set in a Euclidean space contains a closed set of infinite measure, and Prof. Besicovitch asked me whether the closed subset could itself be chosen to be non σ-finite. In this paper an affirmative answer is given.

A functionf:X→ ℝ iscountably decomposable(into continuous functions) if the topological spaceXcan be partitioned into countably many setsAnwith each restrictionf│Ancontinuous. According to L. V. Keldysh(2), the question whether every Baire function is countably decomposable was first raised by N. N. Luzin, and answered by P. S. Novikov. The answer is negative even for Baire-1 functions, as is shown in (2) (see also (1). In this paper we develop a characterization of the countably decomposable functions on a separable metric spaceX(see Corollary 1). We deduce that whenXis complete they include all functions possessing the propertyPdefined by D. E. Peek in (3):each non-empty σ-perfect set H contains a point at which f│ H is continuous. The example given by Peek shows that not every countably decomposable Baire-1 function has propertyP.

An example will be given of a compact linear set K which is of zero or non-σ-finite measure for every translation-invariant Borel measure. This answers a question asked me by C. Dellacherie.

A. J . Goldm an (On measurable se ts and fun c tions, J. Res. NBS 698 (Math. and Math . Phys .) Nos. I and 2, 99100 (1 96.5)) conjectured tha t the Borel se ts are c harac te ri zed by the ir propert y of having meas urable inverse images unde r a ll Le besgue measura ble func tions : he re it is po inted out tha t the e xi s t e nce of analyti c non· Borel sets refut es th is and a re la ted conjeclure. Also an error in Goldm an 's T heore m :2 is correc ted.