We prove that every connected triangle-free graph on n vertices contains an induced tree on exp(c √ log n ) vertices, where c is a positive constant. The best known upper bound is (2 + o(1)) √ n. This partially answers questions of Erdős, Saks, and Sós and of Pultr.

96. Cortassa SC, AonMA, O’Rourke B, Jacques R, Tseng HJ, Marban E, Winslow RL (2006) A computational model integrating electrophysiology, contraction andmitochondrial bioenergetics in the ventricular myocyte. Biophys J 91:1564–1589 97. Saks V, Dzeja P, Schlattner U, Vendelin M, Terzic A, Wallimann T (2006) Cardiac system bioenergetics: metabolic basis of the Frank-Starling law. J Physiol 571:25...

In this paper we investigate the problem of searching monotone multi-dimensional arrays. We generalize Linial and Saks’ search algorithm [2] for monotone 3-dimensional arrays to d-dimensions with d ≥ 4. Our new search algorithm is asymptotically optimal for d = 4.

We prove that the randomized decision tree complexity of the recursive majority-of-three is Ω(2.55), where d is the depth of the recursion. The proof is by a bottom up induction, which is same in spirit as the one in the proof of Saks and Wigderson in their 1986 paper on the complexity of evaluating game trees. Previous work includes an Ω (

Consider a graph obtained by taking edge disjoint union of k complete bipartite graphs. Alon, Saks and Seymour conjectured that such graph has chromatic number at most k + 1. This well known conjecture remained open for almost twenty years. In this paper, we construct a counterexample to this conjecture and discuss several related problems in combinatorial geometry and communication complexity.

The PPSZ Algorithm presented by Paturi, Pudlak, Saks, and Zane in 1998 has the nice feature that the only satisfying solution of a uniquely satisfiable 3-SAT formula can be found in expected running time at most O(1.3071). Using the technique of limited independence, we can derandomize this algorithm yielding O(1.3071) deterministic running time at most.