Abstract. Tuza [ Finite and Infinite Sets, Proc. Colloq. Math. Soc. János Bolyai 37, North Holland, 1981, p. 888] conjectured that [Formula: see text] for all graphs [Formula: see text], where [Formula: see text] is the minimum size of an edge set whose removal makes [Formula: see text] triangle-free and [Formula: see text] is the maximum size of a collection of pairwise edge-disjoint triangles. Here, we generalize Tuza’s conjecture to simple binary matroids that do not contain the Fano plane as a restriction and prove that the geometric version of the conjecture holds for cographic matroids.

A simple binary matroid is called $I_4$-free if none of its rank-4 flats are independent sets. These objects can be equivalently defined as the sets $E$ of points in $\mbox{PG}(n-1,2)$ for which $E \cap F$ is not a basis of $F$ for any four-dimensional flat $F$. We prove a decomposition theorem that exactly determines the structure of all $I_4$-free and triangle-free matroids. In particular, our theorem implies that the $I_4$-free and triangle-free matroids have critical number at most 2.

We applied event-related potential (ERP) to reinforcement signals that are equivalent to reward and punishment signals. We conducted an experiment using an electroencephalogram (EEG) in which volunteers identified the success or failure of an inverted pendulum task. We confirmed that there were differences in the EEG signal depending on whether the task was successful or not and that ERP might be used as a punishment of reinforcement learning. We used a support vector machine (SVM) for recognizing the ERP. We selected the feature vector in SVM that was composed of averages of each 35 msec for each of three channels (F3,Fz,F4) on the frontal area, for a total of 700 msec. Our experimental results suggest that reinforcement learning using ERP can be performed accurately. Finally, we suggest the possibility of developing an adaptive brain-computer interface (BCI) by ERP.