Combinatorics 79 by Peter L. Hammer (Eds.)

By Peter L. Hammer (Eds.)

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For every choice of n, r and k such A general intersection theorem for finite sets that We shall construct a family F of k-subsets of an n-set X such that IF1 a [ n / k l k and yet F does not represent A = {Ic [ 1, r ] 1 111 [ir] + 1). To do so, we partition X into pairwise disjoint subsets XI, X,, . . ,Xk of size at least [n/k] and then set S E F iff )SnXil= 1 for each i. Assuming that F represents A we find members S , , Sz, . . , S, of F such that In particular, for each subset I of [l, r] such that ) I )= [ i r ] + 1 there is a subscript s such that xs n ( si) + 4.

Then a K ” ( A ,B ) is any graph consisting of a complete graph on A, together with a pair of edges connecting each point of B with a different pair of points of A. Furthermore, if F is a K ” ( A ,B ) , define HF to be the graph with A as its vertices, with a pair of vertices joined in HF if they are joined in F through a point of B. Moreover, call a K”(A,B ) in G maximal in a given graph if there exists no K”(A,B , ) in the graph with lBll > (Bl. We will now prove a series of facts about G, leading finally to a contradiction.

3. I1 n’existe pas de snark ayant exactement 16 sommets Un ‘ h a r k ” est un graphe de 2 sans triangle ni carrC, 3 arkte-connexe, cycliquement 4 artte-connexe (l’enlkvement de 3 arktes quelconques ne disconnecte pas le graphe en deux sous-graphes, chacun d’eux contenant un cycle) voir [3]. Le snark Ctait alors considCrC comme Ctant un “animal” mystCrieux et rare, la chasse aux snark prit un Clan nouveau griice a I’article de Isaacs [4] en 1975 qui dCcouvrit une famille infinie de “marks”. Le premier snark connu est naturellement le graphe de Petersen, en fait V n 2 18 nous connaissons au moins un snark d’ordre n ; comme il n’existe pas de snark a I’ordre 12 ni 14, on s’est longtemps demand6 s’il pouvait en exister 2 I’ordre 16: ceci est I’objet de cette partie.

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